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class="vector-toc-list"> <li id="toc-As_partially_ordered_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_partially_ordered_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>As partially ordered set</span> </div> </a> <ul id="toc-As_partially_ordered_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_algebraic_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_algebraic_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>As algebraic structure</span> </div> </a> <ul id="toc-As_algebraic_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_between_the_two_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_between_the_two_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Connection between the two definitions</span> </div> </a> <ul id="toc-Connection_between_the_two_definitions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bounded_lattice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bounded_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Bounded lattice</span> </div> </a> <ul id="toc-Bounded_lattice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_to_other_algebraic_structures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Connection_to_other_algebraic_structures"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Connection to other algebraic structures</span> </div> </a> <ul id="toc-Connection_to_other_algebraic_structures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_non-lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_non-lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples of non-lattices</span> </div> </a> <ul id="toc-Examples_of_non-lattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Morphisms_of_lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Morphisms_of_lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Morphisms of lattices</span> </div> </a> <ul id="toc-Morphisms_of_lattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sublattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sublattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sublattices</span> </div> </a> <ul id="toc-Sublattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_of_lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Properties of lattices</span> </div> </a> <button aria-controls="toc-Properties_of_lattices-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties of lattices subsection</span> </button> <ul id="toc-Properties_of_lattices-sublist" class="vector-toc-list"> <li id="toc-Completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Completeness</span> </div> </a> <ul id="toc-Completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditional_completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Conditional completeness</span> </div> </a> <ul id="toc-Conditional_completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Distributivity</span> </div> </a> <ul id="toc-Distributivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modularity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modularity"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Modularity</span> </div> </a> <ul id="toc-Modularity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semimodularity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semimodularity"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Semimodularity</span> </div> </a> <ul id="toc-Semimodularity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuity_and_algebraicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuity_and_algebraicity"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.6</span> <span>Continuity and algebraicity</span> </div> </a> <ul id="toc-Continuity_and_algebraicity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complements_and_pseudo-complements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complements_and_pseudo-complements"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.7</span> <span>Complements and pseudo-complements</span> </div> </a> <ul id="toc-Complements_and_pseudo-complements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jordan–Dedekind_chain_condition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jordan–Dedekind_chain_condition"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.8</span> <span>Jordan–Dedekind chain condition</span> </div> </a> <ul id="toc-Jordan–Dedekind_chain_condition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graded/ranked" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graded/ranked"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.9</span> <span>Graded/ranked</span> </div> </a> <ul id="toc-Graded/ranked-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Free_lattices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Free_lattices"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Free lattices</span> </div> </a> <ul id="toc-Free_lattices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Important_lattice-theoretic_notions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Important_lattice-theoretic_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Important lattice-theoretic notions</span> </div> </a> <ul id="toc-Important_lattice-theoretic_notions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Applications_that_use_lattice_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications_that_use_lattice_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Applications that use lattice theory</span> </div> </a> <ul id="toc-Applications_that_use_lattice_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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href="https://ca.wikipedia.org/wiki/Reticle_(ordre)" title="Reticle (ordre) – Catalan" lang="ca" hreflang="ca" data-title="Reticle (ordre)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B5%D1%88%D0%B5%D1%82%D0%BA%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Решетке (алгебра) – Chuvash" lang="cv" hreflang="cv" data-title="Решетке (алгебра)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Svaz_(matematika)" title="Svaz (matematika) – Czech" lang="cs" hreflang="cs" data-title="Svaz (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gitter_(ordning)" title="Gitter (ordning) – Danish" lang="da" hreflang="da" data-title="Gitter (ordning)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Verband_(Mathematik)" title="Verband (Mathematik) – German" lang="de" hreflang="de" data-title="Verband (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/V%C3%B5re_(matemaatika)" title="Võre (matemaatika) – Estonian" lang="et" hreflang="et" data-title="Võre (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ret%C3%ADculo_(matem%C3%A1ticas)" title="Retículo (matemáticas) – Spanish" lang="es" hreflang="es" data-title="Retículo (matemáticas)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Latiso_(matematiko)" title="Latiso (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Latiso (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B4%D8%A8%DA%A9%D9%87_(%D8%AA%D8%B1%D8%AA%DB%8C%D8%A8)" title="مشبکه (ترتیب) – Persian" lang="fa" hreflang="fa" data-title="مشبکه (ترتیب)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Treillis_(ensemble_ordonn%C3%A9)" title="Treillis (ensemble ordonné) – French" lang="fr" hreflang="fr" data-title="Treillis (ensemble ordonné)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ret%C3%ADcula_(teor%C3%ADa_da_orde)" title="Retícula (teoría da orde) – Galician" lang="gl" hreflang="gl" data-title="Retícula (teoría da orde)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%A9%EC%9E%90_(%EC%88%9C%EC%84%9C%EB%A1%A0)" title="격자 (순서론) – Korean" lang="ko" hreflang="ko" data-title="격자 (순서론)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BE%D5%A1%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Կավարի տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Կավարի տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kekisi_(tatanan)" title="Kekisi (tatanan) – Indonesian" lang="id" hreflang="id" data-title="Kekisi (tatanan)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Reticulo_(algebra)" title="Reticulo (algebra) – Interlingua" lang="ia" hreflang="ia" data-title="Reticulo (algebra)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Reticolo_(matematica)" title="Reticolo (matematica) – Italian" lang="it" hreflang="it" data-title="Reticolo (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%A8%D7%99%D7%92_(%D7%9E%D7%91%D7%A0%D7%94_%D7%A1%D7%93%D7%95%D7%A8)" title="סריג (מבנה סדור) – Hebrew" lang="he" hreflang="he" data-title="סריג (מבנה סדור)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1l%C3%B3_(matematika)" title="Háló (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Háló (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tralie_(wiskunde)" title="Tralie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Tralie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%9D%9F_(%E6%9D%9F%E8%AB%96)" title="束 (束論) – Japanese" lang="ja" hreflang="ja" data-title="束 (束論)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Ret%C3%ACcol" title="Retìcol – Piedmontese" lang="pms" hreflang="pms" data-title="Retìcol" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Krata_(matematyka)" title="Krata (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Krata (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Reticulado" title="Reticulado – Portuguese" lang="pt" 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searchaux" style="display:none">Set whose pairs have minima and maxima</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice (group)</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output 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style="float:right; margin-top:0; padding-top:0; text-align:center;"> <tbody><tr> <th style="text-align:center;padding-left:0.2em;padding-right:0.2em;font-size:90%;"><span style="font-size:120%"><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a> <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a></span> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output 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navbar-mini" style="float:right; padding-left: 15px; padding-right: 5px;"><ul class="navbar-brackets"><li class="nv-view"><a href="/wiki/Template:Binary_relations" title="Template:Binary relations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Binary_relations" title="Template talk:Binary relations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Binary_relations" title="Special:EditPage/Template:Binary relations"><abbr title="Edit this template">e</abbr></a></li></ul></div> </th></tr> <tr> <td><table style="text-align:center;"><tbody><tr style="vertical-align:middle;"><td style="padding-left:0.3em; padding-right:0.3em;;text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Connected_relation" title="Connected relation">Connected</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Join_and_meet" title="Join and meet">Has joins</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Join_and_meet" title="Join and meet">Has meets</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Reflexive_relation#Irreflexive" title="Reflexive relation">Irreflexive</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"></td><td></td><td></td><td> Total, Semiconnex</td><td></td><td></td><td></td><td></td><td> Anti-<br />reflexive</td><td></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a></span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a href="/wiki/Preorder" title="Preorder">Preorder <span style="font-size:85%;">(Quasiorder)</span></a></span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total preorder</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Total_order" title="Total order">Total order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Prewellordering" title="Prewellordering">Prewellordering</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a></span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Well-order" title="Well-order">Well-ordering</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a class="mw-selflink selflink">Lattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Join-semilattice" class="mw-redirect" title="Join-semilattice">Join-semilattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Meet-semilattice" class="mw-redirect" title="Meet-semilattice">Meet-semilattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict partial order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Weak_ordering#Strict_weak_orderings" title="Weak ordering">Strict weak order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Strict_total_order" class="mw-redirect" title="Strict total order">Strict total order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"></td><td> <a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></td><td> <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></td><td> <a href="/wiki/Connected_relation" title="Connected relation">Connected</a></td><td> <a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></td><td> <a href="/wiki/Join_and_meet" title="Join and meet">Has joins</a></td><td> <a href="/wiki/Join_and_meet" title="Join and meet">Has meets</a></td><td> <a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></td><td> <a href="/wiki/Reflexive_relation#Irreflexive" title="Reflexive relation">Irreflexive</a></td><td> <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> Definitions, for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\neq \varnothing :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\neq \varnothing :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a538fab804c9428c4f7d4ca3ed214a97483c4260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.698ex; height:2.676ex;" alt="{\displaystyle S\neq \varnothing :}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18cac1ed3115c87d45b4d751de57124233a6e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.712ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cc3f46e9f0b647ce6771396a975ef8d364674d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.643ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> <mo stretchy="false">⇒</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c73cadd7ab7bf3b5257f5c505c756ec098902f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.97ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">min</mo> <mi>S</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ec5a418cceeb68b9945ca75d31604719db4660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.513ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3386379a9406067b50b99e12241e9d7fab3369b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.395ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>∧</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abff244e064dd72fd16781277ad3440b35bd767d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.395ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRa}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7fc1d9d50c65105d5edcb3478b5ca4172c54d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.224ex; height:2.176ex;" alt="{\displaystyle aRa}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{not }}aRa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>not </mtext> </mrow> <mi>a</mi> <mi>R</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{not }}aRa}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8263f0c706367e5306eae1b9353034024639da23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.164ex; height:2.176ex;" alt="{\displaystyle {\text{not }}aRa}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mo stretchy="false">⇒</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>not </mtext> </mrow> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aec245f8776556d3ad3fcdc18d4ba61929eccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.683ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}"></span></td></tr></tbody></table> </td></tr> <tr> <td style="text-align:center;"><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> indicates that the column's property is always true for the row's term (at the very left), while <span style="color:red;" title="Red X">✗</span> indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> in the "Symmetric" column and <span style="color:red;" title="Red X">✗</span> in the "Antisymmetric" column, respectively. <br /> <p>All definitions tacitly require the <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> be <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>: for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6021b6ac503535d74098454c2a870a1b5c187d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.949ex; height:2.509ex;" alt="{\displaystyle a,b,c,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRb}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRb}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b0b52168739fd16b254298771ec07b900e5a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.991ex; height:2.176ex;" alt="{\displaystyle aRb}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bRc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>R</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bRc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a62ff0d4c99429cb3465edecb65d2b1a53cf30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.768ex; height:2.176ex;" alt="{\displaystyle bRc}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRc.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e3d14d72455c36aab129326d6dceefafb3f12a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.648ex; height:2.176ex;" alt="{\displaystyle aRc.}"></span> <br /> A term's definition may require additional properties that are not listed in this table. </p> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="display:block;margin-bottom:0.35em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <i><a href="/wiki/Group_theory" title="Group theory">Group theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <i><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a class="mw-selflink selflink">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><i><a class="mw-selflink selflink">Lattice theory</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><i><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> </div></div> <p>A <b>lattice</b> is an abstract structure studied in the <a href="/wiki/Mathematical" class="mw-redirect" title="Mathematical">mathematical</a> subdisciplines of <a href="/wiki/Order_theory" title="Order theory">order theory</a> and <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. It consists of a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> in which every pair of elements has a unique <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> (also called a least upper bound or <a href="/wiki/Join_(mathematics)" class="mw-redirect" title="Join (mathematics)">join</a>) and a unique <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> (also called a greatest lower bound or <a href="/wiki/Meet_(mathematics)" class="mw-redirect" title="Meet (mathematics)">meet</a>). An example is given by the <a href="/wiki/Power_set" title="Power set">power set</a> of a set, partially ordered by <a href="/wiki/Subset" title="Subset">inclusion</a>, for which the supremum is the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> and the infimum is the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>. Another example is given by the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, partially ordered by <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a>, for which the supremum is the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> and the infimum is the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a>. </p><p>Lattices can also be characterized as <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> satisfying certain <a href="/wiki/Axiom" title="Axiom">axiomatic</a> <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a>. Since the two definitions are equivalent, lattice theory draws on both <a href="/wiki/Order_theory" title="Order theory">order theory</a> and <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>. <a href="/wiki/Semilattice" title="Semilattice">Semilattices</a> include lattices, which in turn include <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting</a> and <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a>. These <i>lattice-like</i> structures all admit <a href="/wiki/Order-theoretic" class="mw-redirect" title="Order-theoretic">order-theoretic</a> as well as algebraic descriptions. </p><p>The sub-field of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> that studies lattices is called <b>lattice theory</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure. </p> <div class="mw-heading mw-heading3"><h3 id="As_partially_ordered_set">As partially ordered set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=2" title="Edit section: As partially ordered set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> (poset) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63147107331881c27e29102dfb49f3a745ccccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.234ex; height:2.843ex;" alt="{\displaystyle (L,\leq )}"></span> is called a <b>lattice</b> if it is both a join- and a meet-<a href="/wiki/Semilattice" title="Semilattice">semilattice</a>, i.e. each two-element subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b\}\subseteq L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b\}\subseteq L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbaf609fbfedc5d2e34cb3b8b50a164bb7124685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.268ex; height:2.843ex;" alt="{\displaystyle \{a,b\}\subseteq L}"></span> has a <a href="/wiki/Join_(mathematics)" class="mw-redirect" title="Join (mathematics)">join</a> (i.e. least upper bound, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3393da6721f85fa89a1d3a8c28e82c679abe4032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\vee b}"></span>) and <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dually</a> a <a href="/wiki/Meet_(mathematics)" class="mw-redirect" title="Meet (mathematics)">meet</a> (i.e. greatest lower bound, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc496a5b5da3e9b94eb72f04a54167dfe022e45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\wedge b}"></span>). This definition makes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\wedge \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∧</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\wedge \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7dc9daaf40e757637bdf320adcebb3be73c7b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\wedge \,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\vee \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∨</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\vee \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234a83a87e5dbd07cd5793fb463be0e7995a01f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.009ex;" alt="{\displaystyle \,\vee \,}"></span> <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a>. Both operations are monotone with respect to the given order: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\leq a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\leq a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e46279da35360183960458a1fb9fee454e2583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.343ex;" alt="{\displaystyle a_{1}\leq a_{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}\leq b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}\leq b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51153250b2e0350954f15e9b83f147067c66838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.202ex; height:2.509ex;" alt="{\displaystyle b_{1}\leq b_{2}}"></span> implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\vee b_{1}\leq a_{2}\vee b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\vee b_{1}\leq a_{2}\vee b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4206627b2083e877e4707b9537da0a508bead6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.935ex; height:2.509ex;" alt="{\displaystyle a_{1}\vee b_{1}\leq a_{2}\vee b_{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afbefc55f4bcc6d1b2982e4134eaaf94fbda9cf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.582ex; height:2.509ex;" alt="{\displaystyle a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.}"></span> </p><p>It follows by an <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; see <i><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness (order theory)</a></i> for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable <a href="/wiki/Galois_connection" title="Galois connection">Galois connections</a> between related partially ordered sets—an approach of special interest for the <a href="/wiki/Category_theoretic" class="mw-redirect" title="Category theoretic">category theoretic</a> approach to lattices, and for <a href="/wiki/Formal_concept_analysis" title="Formal concept analysis">formal concept analysis</a>. </p><p>Given a subset of a lattice, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subseteq L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77d9d73b8e791b68ab3fe9489b027c0c5ee5d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.392ex; height:2.509ex;" alt="{\displaystyle H\subseteq L,}"></span> meet and join restrict to <a href="/wiki/Partial_function" title="Partial function">partial functions</a> – they are undefined if their value is not in the subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8933ae7244305ae7824aa18e077d1cf946e2ee9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.71ex; height:2.176ex;" alt="{\displaystyle H.}"></span> The resulting structure on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is called a <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="partial_lattice"></span><span class="vanchor-text">partial lattice</span></span></b>. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.<sup id="cite_ref-FOOTNOTEGrätzer2003[httpsbooksgooglecombooksidSoGLVCPuOz0CpgPA52_52]_1-0" class="reference"><a href="#cite_note-FOOTNOTEGrätzer2003[httpsbooksgooglecombooksidSoGLVCPuOz0CpgPA52_52]-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="As_algebraic_structure">As algebraic structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=3" title="Edit section: As algebraic structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>lattice</b> is an <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f941695c9d00a4185f950e8f9e7e50e28d2fcf3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge )}"></span>, consisting of a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> and two binary, commutative and associative <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> satisfying the following axiomatic identities for all elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41807417fb4f6dd64a2bbb60b5a9132a7e803782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.685ex; height:2.509ex;" alt="{\displaystyle a,b\in L}"></span> (sometimes called <em>absorption laws</em>): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee (a\wedge b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee (a\wedge b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d56d38af00002b148730cc0ff4218d81a3b884" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.76ex; height:2.843ex;" alt="{\displaystyle a\vee (a\wedge b)=a}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge (a\vee b)=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge (a\vee b)=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0caa61ea861e816dfb222fc78f5733cc925008" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.76ex; height:2.843ex;" alt="{\displaystyle a\wedge (a\vee b)=a}"></span> </p><p>The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> These are called <em>idempotent laws</em>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178afc08d2ea4985bceff5f230c55d637f5d46a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.37ex; height:2.009ex;" alt="{\displaystyle a\vee a=a}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca87f433c2782f31140e20d9516a737a36e68c5d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.37ex; height:2.009ex;" alt="{\displaystyle a\wedge a=a}"></span> </p><p>These axioms assert that both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe0334d6d981cac827ed35a839df8fc14ae59a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.976ex; height:2.843ex;" alt="{\displaystyle (L,\vee )}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64199785cca493a1daabff28aa5e222df68fbaff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.976ex; height:2.843ex;" alt="{\displaystyle (L,\wedge )}"></span> are <a href="/wiki/Semilattice" title="Semilattice">semilattices</a>. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual</a> of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Connection_between_the_two_definitions">Connection between the two definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=4" title="Edit section: Connection between the two definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An order-theoretic lattice gives rise to the two binary operations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d52762765c839189ca0de462ae0c350f2e68401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.009ex;" alt="{\displaystyle \wedge .}"></span> Since the commutative, associative and absorption laws can easily be verified for these operations, they make <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f941695c9d00a4185f950e8f9e7e50e28d2fcf3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge )}"></span> into a lattice in the algebraic sense. </p><p>The converse is also true. Given an algebraically defined lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7afb51334a5bf7fc2063a2de1eeaac609b88dfbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.208ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge ),}"></span> one can define a partial order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> by setting <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6dba8beddf1ac365e7a9bf647fa104fa3d39404" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.253ex; height:2.509ex;" alt="{\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b{\text{ if }}b=a\vee b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b{\text{ if }}b=a\vee b,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3bc75581be4bd1c43a186a8dc7fc2f15932563" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.399ex; height:2.509ex;" alt="{\displaystyle a\leq b{\text{ if }}b=a\vee b,}"></span> for all elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29aae3d806a3ec4ce45defc7365a525515971b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.332ex; height:2.509ex;" alt="{\displaystyle a,b\in L.}"></span> The laws of absorption ensure that both definitions are equivalent: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> implies </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaecfbf6f3b2c2e6f1f073cfd2a0de8d9a8285cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.919ex; height:2.843ex;" alt="{\displaystyle a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b}"></span> and dually for the other direction. </p><p>One can now check that the relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span> introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d52762765c839189ca0de462ae0c350f2e68401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.009ex;" alt="{\displaystyle \wedge .}"></span> </p><p>Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand. </p> <div class="mw-heading mw-heading2"><h2 id="Bounded_lattice">Bounded lattice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=5" title="Edit section: Bounded lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>bounded lattice</b> is a lattice that additionally has a <i><a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element"><dfn>greatest element</dfn></a></i> (also called <i><dfn>maximum</dfn></i>, or <i><dfn>top</dfn></i> element, and denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"></span> or <span class="nowrap">by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"></span>)</span> and a <i><a href="/wiki/Least_element" class="mw-redirect" title="Least element"><dfn>least element</dfn></a></i> (also called <i><dfn>minimum</dfn></i>, or <i><dfn>bottom</dfn></i>, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> or by <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></span>),</span> which satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1\;{\text{ for every }}x\in L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for every </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1\;{\text{ for every }}x\in L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5f643756c50b72bb1d41d6e1408ec23c2b55ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.856ex; height:2.509ex;" alt="{\displaystyle 0\leq x\leq 1\;{\text{ for every }}x\in L.}"></span> </p><p>A bounded lattice may also be defined as an algebraic structure of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge ,0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge ,0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ab00d1693d5c620aa0f87e460b8d02bcfa8e219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.953ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge ,0,1)}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f941695c9d00a4185f950e8f9e7e50e28d2fcf3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge )}"></span> is a lattice, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> (the lattice's bottom) is the <a href="/wiki/Identity_element" title="Identity element">identity element</a> for the join operation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/336baf7039048ca8199bfa51dca9da5c874c2786" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.343ex;" alt="{\displaystyle \vee ,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> (the lattice's top) is the identity element for the meet operation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d52762765c839189ca0de462ae0c350f2e68401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.009ex;" alt="{\displaystyle \wedge .}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee 0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee 0=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e396f7856e78165c287e30ad68223b3f9dad692" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.303ex; height:2.176ex;" alt="{\displaystyle a\vee 0=a}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge 1=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge 1=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342bb18c985f0ce6a6dc319454d6ae0178374712" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.303ex; height:2.176ex;" alt="{\displaystyle a\wedge 1=a}"></span> </p><p>A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of a poset it is <a href="/wiki/Vacuously_true" class="mw-redirect" title="Vacuously true">vacuously true</a> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>a</mi> <mo>∈</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mi>x</mi> <mo>≤</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79acfbed7cf5ee841590f33277cfa6586a3b5b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.555ex; height:2.509ex;" alt="{\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{ for all }}a\in \varnothing ,a\leq x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>a</mi> <mo>∈</mo> <mi class="MJX-variant">∅</mi> <mo>,</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ for all }}a\in \varnothing ,a\leq x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc7ece77ca4102b8c2650787c1c3163301a9928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.201ex; height:2.509ex;" alt="{\displaystyle {\text{ for all }}a\in \varnothing ,a\leq x,}"></span> and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigvee \varnothing =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋁</mo> <mi class="MJX-variant">∅</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigvee \varnothing =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77b6ee67ce79db0d3fde1e8fff80ba9257f3c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.039ex; height:2.843ex;" alt="{\textstyle \bigvee \varnothing =0,}"></span> and the meet of the empty set is the greatest element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigwedge \varnothing =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>⋀</mo> <mi class="MJX-variant">∅</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigwedge \varnothing =1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833ace8a830c6b72e0afb28c744a8312d787999c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.039ex; height:2.843ex;" alt="{\textstyle \bigwedge \varnothing =1.}"></span> This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, that is, for finite subsets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> of a poset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋁</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∨</mo> <mrow> <mo>(</mo> <mrow> <mo>⋁</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a83a360c3531b218a01cd1988026de6ed51ed1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.157ex; height:4.843ex;" alt="{\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋀</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋀</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∧</mo> <mrow> <mo>(</mo> <mrow> <mo>⋀</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2185827dc2450b9c9782eaf8e00b85520fc8a514" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.157ex; height:4.843ex;" alt="{\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}"></span> hold. Taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> to be the empty set, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋁</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∨</mo> <mrow> <mo>(</mo> <mrow> <mo>⋁</mo> <mi class="MJX-variant">∅</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋁</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∨</mo> <mn>0</mn> <mo>=</mo> <mo>⋁</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afa8bec87ad29bba8374ef9fe0bcc60516a0f0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.386ex; height:4.843ex;" alt="{\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋀</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋀</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∧</mo> <mrow> <mo>(</mo> <mrow> <mo>⋀</mo> <mi class="MJX-variant">∅</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>⋀</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>∧</mo> <mn>1</mn> <mo>=</mo> <mo>⋀</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a01c07e936d212c3e61582ed5bdc7be719818d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.033ex; height:4.843ex;" alt="{\displaystyle \bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A,}"></span> which is consistent with the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cup \varnothing =A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∪</mo> <mi class="MJX-variant">∅</mi> <mo>=</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cup \varnothing =A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed098975c23f01b40c364ec076e575545feec39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.622ex; height:2.176ex;" alt="{\displaystyle A\cup \varnothing =A.}"></span> </p><p>Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mo>⋁</mo> <mi>L</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <mo>⋯</mo> <mo>∨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709eab387014b98a464b84d42ac021b203934bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.886ex; height:2.843ex;" alt="{\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}"></span> (respectively <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mo>⋀</mo> <mi>L</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1c3472166da6d552201196573cd869eee6dc88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.886ex; height:2.843ex;" alt="{\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}"></span>) where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9beaa251841ab11245eda68877586bdf0e9bc56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.917ex; height:2.843ex;" alt="{\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}"></span> is the set of all elements. </p> <div class="mw-heading mw-heading2"><h2 id="Connection_to_other_algebraic_structures">Connection to other algebraic structures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=6" title="Edit section: Connection to other algebraic structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lattices have some connections to the family of <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">group-like algebraic structures</a>. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative <a href="/wiki/Semigroups" class="mw-redirect" title="Semigroups">semigroups</a> having the same domain. For a bounded lattice, these semigroups are in fact commutative <a href="/wiki/Monoid" title="Monoid">monoids</a>. The <a href="/wiki/Absorption_law" title="Absorption law">absorption law</a> is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative <a href="/wiki/Rig_(mathematics)" class="mw-redirect" title="Rig (mathematics)">rig</a> without the distributive axiom. </p><p>By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"></span> respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded. </p><p>The algebraic interpretation of lattices plays an essential role in <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="The whole section (as far as it sentences convey information rather than opinion) should be sourced. (January 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Hasse_diagram_of_powerset_of_3.svg" class="mw-file-description" title="Pic. 1: Subsets of '"`UNIQ--postMath-00000057-QINU`"' under set inclusion. The name "lattice" is suggested by the form of the Hasse diagram depicting it."><img alt="Pic. 1: Subsets of '"`UNIQ--postMath-00000057-QINU`"' under set inclusion. The name "lattice" is suggested by the form of the Hasse diagram depicting it." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/120px-Hasse_diagram_of_powerset_of_3.svg.png" decoding="async" width="120" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/180px-Hasse_diagram_of_powerset_of_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hasse_diagram_of_powerset_of_3.svg/240px-Hasse_diagram_of_powerset_of_3.svg.png 2x" data-file-width="429" data-file-height="325" /></a></span></div> <div class="gallerytext"><b>Pic. 1:</b> Subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y,z\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,y,z\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5738613a098266a2028c572b89d028f5f40a9736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.613ex; height:2.843ex;" alt="{\displaystyle \{x,y,z\},}"></span> under <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a>. The name "lattice" is suggested by the form of the <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> depicting it.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Pic. 2: Lattice of integer divisors of 60, ordered by "divides"."><img alt="Pic. 2: Lattice of integer divisors of 60, ordered by "divides"." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/120px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/180px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/240px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div> <div class="gallerytext"><b>Pic. 2:</b> Lattice of integer divisors of 60, ordered by "<i>divides</i>".</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Lattice_of_partitions_of_an_order_4_set.svg" class="mw-file-description" title="Pic. 3: Lattice of partitions of '"`UNIQ--postMath-00000058-QINU`"' ordered by "refines"."><img alt="Pic. 3: Lattice of partitions of '"`UNIQ--postMath-00000058-QINU`"' ordered by "refines"." src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Lattice_of_partitions_of_an_order_4_set.svg/118px-Lattice_of_partitions_of_an_order_4_set.svg.png" decoding="async" width="118" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Lattice_of_partitions_of_an_order_4_set.svg/178px-Lattice_of_partitions_of_an_order_4_set.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Lattice_of_partitions_of_an_order_4_set.svg/237px-Lattice_of_partitions_of_an_order_4_set.svg.png 2x" data-file-width="289" data-file-height="293" /></a></span></div> <div class="gallerytext"><b>Pic. 3:</b> Lattice of <a href="/wiki/Partition_(set_theory)" class="mw-redirect" title="Partition (set theory)">partitions</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb89b8ef513b1960954e9de58b86ea8b1072754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.723ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4\},}"></span> ordered by "<i>refines</i>".</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Nat_num.svg" class="mw-file-description" title="Pic. 4: Lattice of positive integers, ordered by '"`UNIQ--postMath-00000059-QINU`"'"><img alt="Pic. 4: Lattice of positive integers, ordered by '"`UNIQ--postMath-00000059-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/120px-Nat_num.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/180px-Nat_num.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Nat_num.svg/240px-Nat_num.svg.png 2x" data-file-width="120" data-file-height="120" /></a></span></div> <div class="gallerytext"><b>Pic. 4:</b> Lattice of positive integers, ordered by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\leq ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>≤</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\leq ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5297c6496868eb8fd01d1a33480043d1d3ffdf73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.842ex; height:2.343ex;" alt="{\displaystyle \,\leq ,}"></span></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:N-Quadrat,_gedreht.svg" class="mw-file-description" title="Pic. 5: Lattice of nonnegative integer pairs, ordered componentwise."><img alt="Pic. 5: Lattice of nonnegative integer pairs, ordered componentwise." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/120px-N-Quadrat%2C_gedreht.svg.png" decoding="async" width="120" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/180px-N-Quadrat%2C_gedreht.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/240px-N-Quadrat%2C_gedreht.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span></div> <div class="gallerytext"><b>Pic. 5:</b> Lattice of nonnegative integer pairs, ordered componentwise.</div> </li> </ul> <ul><li>For any set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> the collection of all subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (called the <a href="/wiki/Power_set" title="Power set">power set</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>) can be ordered via <a href="/wiki/Subset_inclusion" class="mw-redirect" title="Subset inclusion">subset inclusion</a> to obtain a lattice bounded by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> itself and the empty set. In this lattice, the supremum is provided by <a href="/wiki/Set_union" class="mw-redirect" title="Set union">set union</a> and the infimum is provided by <a href="/wiki/Set_intersection" class="mw-redirect" title="Set intersection">set intersection</a> (see Pic. 1).</li> <li>For any set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> the collection of all finite subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> ordered by inclusion, is also a lattice, and will be bounded if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is finite.</li> <li>For any set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> the collection of all <a href="/wiki/Partition_of_a_set" title="Partition of a set">partitions</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> ordered by <a href="/wiki/Partition_of_a_set" title="Partition of a set">refinement</a>, is a lattice (see Pic. 3).</li> <li>The <a href="/wiki/Positive_integers" class="mw-redirect" title="Positive integers">positive integers</a> in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4).</li> <li>The <a href="/wiki/Cartesian_square" class="mw-redirect" title="Cartesian square">Cartesian square</a> of the natural numbers, ordered so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\leq (c,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\leq (c,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46430c9eaa44bf1a6e11bc8afc88e0bdedeea83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.235ex; height:2.843ex;" alt="{\displaystyle (a,b)\leq (c,d)}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq c{\text{ and }}b\leq d.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>b</mi> <mo>≤</mo> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq c{\text{ and }}b\leq d.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8fbb5dd82e3d3194f0b42dda1f26e949518c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.203ex; height:2.343ex;" alt="{\displaystyle a\leq c{\text{ and }}b\leq d.}"></span> The pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> is the bottom element; there is no top (see Pic. 5).</li> <li>The natural numbers also form a lattice under the operations of taking the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> and <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a>, with <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a> as the order relation: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> divides <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef051eb30c89e5493d672f6479566c673b0890a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> is bottom; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> is top. Pic. 2 shows a finite sublattice.</li> <li>Every <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> (also see <a href="#Completeness">below</a>) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical <a href="/wiki/Complete_lattice#Examples" title="Complete lattice">examples</a>.</li> <li>The set of <a href="/wiki/Compact_element" title="Compact element">compact elements</a> of an <a href="/wiki/Arithmetic_lattice" class="mw-redirect" title="Arithmetic lattice">arithmetic</a> complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from <a href="/wiki/Algebraic_lattice" class="mw-redirect" title="Algebraic lattice">algebraic lattices</a>, for which the compacts only form a <a href="/wiki/Join-semilattice" class="mw-redirect" title="Join-semilattice">join-semilattice</a>. Both of these classes of complete lattices are studied in <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>.</li></ul> <p>Further examples of lattices are given for each of the additional properties discussed below. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Examples_of_non-lattices">Examples of non-lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=8" title="Edit section: Examples of non-lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pow3nonlattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Pow3nonlattice.svg/155px-Pow3nonlattice.svg.png" decoding="async" width="155" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Pow3nonlattice.svg/233px-Pow3nonlattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Pow3nonlattice.svg/310px-Pow3nonlattice.svg.png 2x" data-file-width="429" data-file-height="415" /></a><figcaption><b>Pic. 8:</b> Non-lattice poset: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> have common lower bounds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,d,g,h,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,d,g,h,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/263623a77de741bf881d6cfdebb58e0c9fbb2f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.582ex; height:2.509ex;" alt="{\displaystyle 0,d,g,h,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d0f7dadba3056fa3c06a6bee5c0b4182471152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle i,}"></span> but none of them is the <a href="/wiki/Greatest_lower_bound" class="mw-redirect" title="Greatest lower bound">greatest lower bound</a>.</figcaption></figure> </td></tr></tbody></table> <table style="float:right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NoLatticeDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/NoLatticeDiagram.svg/177px-NoLatticeDiagram.svg.png" decoding="async" width="177" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/NoLatticeDiagram.svg/266px-NoLatticeDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/NoLatticeDiagram.svg/355px-NoLatticeDiagram.svg.png 2x" data-file-width="488" data-file-height="413" /></a><figcaption><b>Pic. 7:</b> Non-lattice poset: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> have common upper bounds <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d,e,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d,e,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d841308beaba598133100b8180504b1d3909f7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.98ex; height:2.509ex;" alt="{\displaystyle d,e,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}"></span> but none of them is the <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a>.</figcaption></figure> </td></tr></tbody></table> <table style="float:right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:KeinVerband.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/KeinVerband.svg/150px-KeinVerband.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/KeinVerband.svg/225px-KeinVerband.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/KeinVerband.svg/300px-KeinVerband.svg.png 2x" data-file-width="80" data-file-height="80" /></a><figcaption><b>Pic. 6:</b> Non-lattice poset: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> have no common upper bound.</figcaption></figure> </td></tr></tbody></table> <p>Most partially ordered sets are not lattices, including the following. </p> <ul><li>A discrete poset, meaning a poset such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30afb48bff6ee3b369304f64314cbec0aca3798d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.009ex;" alt="{\displaystyle x=y,}"></span> is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.</li> <li>Although the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,6\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,6\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9695f2cd87f42d9c261cfd6a95b312b78ec42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.077ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,6\}}"></span> partially ordered by divisibility is a lattice, the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span> so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{2,3,6\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{2,3,6\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8506dde8628653e8bede60072bb6c21fee111b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle \{2,3,6\}.}"></span></li> <li>The set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,12,18,36\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>36</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3,12,18,36\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b94e2da411913152824b4faa6e7bc5f7c30a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.957ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,12,18,36\}}"></span> partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).</li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Morphisms_of_lattices">Morphisms of lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=9" title="Edit section: Morphisms of lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Monotonic_but_nonhomomorphic_map_between_lattices.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif/220px-Monotonic_but_nonhomomorphic_map_between_lattices.gif" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif/330px-Monotonic_but_nonhomomorphic_map_between_lattices.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Monotonic_but_nonhomomorphic_map_between_lattices.gif/440px-Monotonic_but_nonhomomorphic_map_between_lattices.gif 2x" data-file-width="714" data-file-height="400" /></a><figcaption><b>Pic. 9:</b> Monotonic map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> between lattices that preserves neither joins nor meets, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>∨</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc24379d650bd4bffa3c2638becae1e8e7b26e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.038ex; height:3.009ex;" alt="{\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cc3d8d8c60120bc2f905bae4d5e10d8ad6a3f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \neq }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{\prime }=f(1)=f(u\vee v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>∨</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{\prime }=f(1)=f(u\vee v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a516742e51937a38bddd7c48a5b06869318de16b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.422ex; height:3.009ex;" alt="{\displaystyle 1^{\prime }=f(1)=f(u\vee v)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>∧</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51745433ab9513edeb81016c4886c822d421e98d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.038ex; height:3.009ex;" alt="{\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≠</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cc3d8d8c60120bc2f905bae4d5e10d8ad6a3f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.808ex; height:2.676ex;" alt="{\displaystyle \neq }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>∧</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7e110bc62c4538124375ca9c1a3a8d88690edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.069ex; height:3.009ex;" alt="{\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}"></span></figcaption></figure> <p>The appropriate notion of a <a href="/wiki/Morphism" title="Morphism">morphism</a> between two lattices flows easily from the <a href="#Lattices_as_algebraic_structures">above</a> algebraic definition. Given two lattices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>,</mo> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ea379958de351303f43a0762ead51f122bee59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.264ex; height:2.843ex;" alt="{\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mo>,</mo> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1e95c957c17a121c474097cce4b5aaf45e8122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.372ex; height:2.843ex;" alt="{\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}"></span> a <b>lattice homomorphism</b> from <i>L</i> to <i>M</i> is a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:L\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>L</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:L\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aafb59f5ff0d174782c246bb8cab971b57ed6cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.855ex; height:2.509ex;" alt="{\displaystyle f:L\to M}"></span> such that for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in L:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>L</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in L:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cc1f4fa079138a604aab23f7ae4b872abc9801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.977ex; height:2.509ex;" alt="{\displaystyle a,b\in L:}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a08394ddac87345dc8cc70a644029f2e7b672d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.622ex; height:2.843ex;" alt="{\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df1e942aa6d313c2471f8ab98325abf632ca67a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.327ex; height:2.843ex;" alt="{\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}"></span> </p><p>Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is a <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of the two underlying <a href="/wiki/Semilattice" title="Semilattice">semilattices</a>. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a <b>bounded-lattice homomorphism</b> (usually called just "lattice homomorphism") <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> between two bounded lattices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> should also have the following property: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a48417c3cb18a611303dc61108776059e681d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.152ex; height:2.843ex;" alt="{\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(1_{L}\right)=1_{M}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(1_{L}\right)=1_{M}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab528d9c19e8761f4cca8cad446baac2e77e8b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.856ex; height:2.843ex;" alt="{\displaystyle f\left(1_{L}\right)=1_{M}.}"></span> </p><p>In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function <a href="/wiki/Limit_preserving_function_(order_theory)" class="mw-redirect" title="Limit preserving function (order theory)">preserving</a> binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set. </p><p>Any homomorphism of lattices is necessarily <a href="/wiki/Monotone_function" class="mw-redirect" title="Monotone function">monotone</a> with respect to the associated ordering relation; see <a href="/wiki/Limit_preserving_function_(order_theory)" class="mw-redirect" title="Limit preserving function (order theory)">Limit preserving function</a>. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an <a href="/wiki/Monotonic_function" title="Monotonic function">order-preserving</a> <a href="/wiki/Bijection" title="Bijection">bijection</a> is a homomorphism if its <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> is also order-preserving. </p><p>Given the standard definition of <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a> as invertible morphisms, a <i><dfn>lattice isomorphism</dfn></i> is just a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> lattice homomorphism. Similarly, a <i><dfn>lattice endomorphism</dfn></i> is a lattice homomorphism from a lattice to itself, and a <i><dfn>lattice automorphism</dfn></i> is a bijective lattice endomorphism. Lattices and their homomorphisms form a <a href="/wiki/Category_theory" title="Category theory">category</a>. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {L} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2f600f4d0a100f870d96a8409cdf4af65c592f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.235ex; height:2.509ex;" alt="{\displaystyle \mathbb {L} '}"></span> be two lattices with <b>0</b> and <b>1</b>. A homomorphism from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c54e1ea2df1f3d345e5ecea9313712f999d3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {L} }"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {L} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">L</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {L} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2f600f4d0a100f870d96a8409cdf4af65c592f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.235ex; height:2.509ex;" alt="{\displaystyle \mathbb {L} '}"></span> is called <b>0</b>,<b>1</b>-<i>separating</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\{f(0)\}=\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\{f(0)\}=\{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0098b69fbb402e4a4ce52d4c72755baeb1c75d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.814ex; height:3.176ex;" alt="{\displaystyle f^{-1}\{f(0)\}=\{0\}}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> separates <b>0</b>) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\{f(1)\}=\{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\{f(1)\}=\{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7569966a3fec15bc47c58857ffa2849d3a5f3634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.814ex; height:3.176ex;" alt="{\displaystyle f^{-1}\{f(1)\}=\{1\}}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> separates 1). </p> <div class="mw-heading mw-heading2"><h2 id="Sublattices">Sublattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=10" title="Edit section: Sublattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i><dfn>sublattice</dfn></i> of a lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> that is a lattice with the same meet and join operations as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6358b7808f1224473a219f51e5eede2495fa88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle L.}"></span> That is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is a lattice and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> such that for every pair of elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb00e3fe8849131890d2b4db8373457573bd0a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:2.509ex;" alt="{\displaystyle a,b\in M}"></span> both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc496a5b5da3e9b94eb72f04a54167dfe022e45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\wedge b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3393da6721f85fa89a1d3a8c28e82c679abe4032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\vee b}"></span> are in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b466e90209f39c0c2caad1b11445824b82c2f536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a sublattice of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6358b7808f1224473a219f51e5eede2495fa88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle L.}"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A sublattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> of a lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is a <em>convex sublattice</em> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq z\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>z</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq z\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc14a54fe48ebc7a2729afba14cc619270b0f0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.343ex;" alt="{\displaystyle x\leq z\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea304ca242a255b620d3dd16ec47f19efc2e7ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\displaystyle x,y\in M}"></span> implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> belongs to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b466e90209f39c0c2caad1b11445824b82c2f536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}"></span> for all elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d683a4f1db36838d3fe9fdc0028e12b0d696708e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.712ex; height:2.509ex;" alt="{\displaystyle x,y,z\in L.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Properties_of_lattices">Properties of lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=11" title="Edit section: Properties of lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></div> <p>We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed. </p> <div class="mw-heading mw-heading3"><h3 id="Completeness">Completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=12" title="Edit section: Completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complete_lattice" title="Complete lattice">Complete lattice</a></div> <p>A poset is called a <i><dfn>complete lattice</dfn></i> if <em>all</em> its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets. </p><p>Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices. </p><p>"Partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions. </p> <div class="mw-heading mw-heading3"><h3 id="Conditional_completeness">Conditional completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=13" title="Edit section: Conditional completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dedekind_complete" class="mw-redirect" title="Dedekind complete">Dedekind complete</a></div> <p>A <b>conditionally complete lattice</b> is a lattice in which every <em>nonempty</em> subset <em>that has an upper bound</em> has a join (that is, a least upper bound). Such lattices provide the most direct generalization of the <a href="/wiki/Completeness_axiom" class="mw-redirect" title="Completeness axiom">completeness axiom</a> of the <a href="/wiki/Real_number" title="Real number">real numbers</a>. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"></span> its minimum element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95547343453ea34a314dd174f8458012f5a39ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0,}"></span> or both.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Distributivity">Distributivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=14" title="Edit section: Distributivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:N_5_mit_Beschriftung.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/N_5_mit_Beschriftung.svg/150px-N_5_mit_Beschriftung.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/N_5_mit_Beschriftung.svg/225px-N_5_mit_Beschriftung.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/N_5_mit_Beschriftung.svg/300px-N_5_mit_Beschriftung.svg.png 2x" data-file-width="180" data-file-height="180" /></a><figcaption><b>Pic. 11:</b> Smallest non-modular (and hence non-distributive) lattice N<sub>5</sub>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≤</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\leq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04cbc237b132cef779abc512c9c8e288781a808e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.103ex; height:2.343ex;" alt="{\displaystyle b\leq c}"></span>, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\vee (a\wedge c)=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\vee (a\wedge c)=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/544789b4f89fdc7ad9bc62db8430e5617c68efd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.305ex; height:2.843ex;" alt="{\displaystyle b\vee (a\wedge c)=b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b\vee a)\wedge c=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∨</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>c</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b\vee a)\wedge c=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/962bb00ec95965c7582644cae54a393e7edace42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.314ex; height:2.843ex;" alt="{\displaystyle (b\vee a)\wedge c=c}"></span>, so the modular law is violated.<br />The labelled elements also violate the distributivity equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>∧</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a893dafbbd056195e14c59db3af68f597fb536c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.561ex; height:2.843ex;" alt="{\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),}"></span> but satisfy its dual <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>∨</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b307093671d4497f5eb10a759ed7d772c4adfba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.561ex; height:2.843ex;" alt="{\displaystyle c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).}"></span></figcaption></figure> </td></tr></tbody></table> <table style="float:right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:M_3_mit_Beschriftung.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/M_3_mit_Beschriftung.svg/150px-M_3_mit_Beschriftung.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/M_3_mit_Beschriftung.svg/225px-M_3_mit_Beschriftung.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/M_3_mit_Beschriftung.svg/300px-M_3_mit_Beschriftung.svg.png 2x" data-file-width="120" data-file-height="120" /></a><figcaption><b>Pic. 10:</b> Smallest non-distributive (but modular) lattice M<sub>3</sub>.</figcaption></figure> </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive lattice</a></div> <p>Since lattices come with two binary operations, it is natural to ask whether one of them <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributes</a> over the other, that is, whether one or the other of the following <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual</a> laws holds for every three elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbd29e4923b241b11751a355ba714c5a301795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.372ex; height:2.509ex;" alt="{\displaystyle a,b,c\in L,}"></span>: </p> <dl><dt>Distributivity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span></dt></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b15fc9fcf5fb9faa2d6e1c0b7baff6458a9461b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.784ex; height:2.843ex;" alt="{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}"></span> </p> <dl><dt>Distributivity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span></dt></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∨</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970a8d0e9d7f001f99badb3ee76a4919f9638111" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.784ex; height:2.843ex;" alt="{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).}"></span> </p><p>A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a <b>distributive lattice</b>. The only non-distributive lattices with fewer than 6 elements are called M<sub>3</sub> and N<sub>5</sub>;<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a <a href="#Sublattices">sublattice</a> isomorphic to M<sub>3</sub> or N<sub>5</sub>.<sup id="cite_ref-Davey.Priestley.2002.10.6_7-0" class="reference"><a href="#cite_note-Davey.Priestley.2002.10.6-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as <a href="/wiki/Complete_Heyting_algebra" title="Complete Heyting algebra">frames</a> and <a href="/wiki/Completely_distributive_lattice" title="Completely distributive lattice">completely distributive lattices</a>, see <a href="/wiki/Distributivity_(order_theory)" title="Distributivity (order theory)">distributivity in order theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Modularity">Modularity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=15" title="Edit section: Modularity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Modular_lattice" title="Modular lattice">Modular lattice</a></div> <p>For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\vee ,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\vee ,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f941695c9d00a4185f950e8f9e7e50e28d2fcf3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle (L,\vee ,\wedge )}"></span> is <i><dfn>modular</dfn></i> if, for all elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbd29e4923b241b11751a355ba714c5a301795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.372ex; height:2.509ex;" alt="{\displaystyle a,b,c\in L,}"></span> the following identity holds: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3aacbb298ff507c2b341043661af38f4982cc0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.96ex; height:2.843ex;" alt="{\displaystyle (a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.}"></span> (<dfn>Modular identity</dfn>)<br /> This condition is equivalent to the following axiom: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1c962997d8a303e076777cd6d6bc732f360ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.335ex; height:2.176ex;" alt="{\displaystyle a\leq c}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∧</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be641ee745405be3e97df02ddd72d693b14292dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.163ex; height:2.843ex;" alt="{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge c.}"></span> (<dfn>Modular law</dfn>)<br /> A lattice is modular if and only if it does not have a <a href="/wiki/Sublattice" class="mw-redirect" title="Sublattice">sublattice</a> isomorphic to N<sub>5</sub> (shown in Pic. 11).<sup id="cite_ref-Davey.Priestley.2002.10.6_7-1" class="reference"><a href="#cite_note-Davey.Priestley.2002.10.6-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Besides distributive lattices, examples of modular lattices are the lattice of submodules of a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> (hence <i>modular</i>), the lattice of <a href="/wiki/Two-sided_ideal" class="mw-redirect" title="Two-sided ideal">two-sided ideals</a> of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and the lattice of <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. The <a href="/wiki/Subsumption_lattice" title="Subsumption lattice">set of first-order terms</a> with the ordering "is more specific than" is a non-modular lattice used in <a href="/wiki/Automated_reasoning" title="Automated reasoning">automated reasoning</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Semimodularity">Semimodularity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=16" title="Edit section: Semimodularity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Semimodular_lattice" title="Semimodular lattice">Semimodular lattice</a></div> <p>A finite lattice is modular if and only if it is both upper and lower <a href="/wiki/Semimodular_lattice" title="Semimodular lattice">semimodular</a>. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bef0cfcf31f9f7244bf041657e5e68ecbac537d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.695ex; height:1.676ex;" alt="{\displaystyle r\colon }"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∨</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39863337e8a2b836576fe0449846d1eadf265d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.479ex; height:2.843ex;" alt="{\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).}"></span></dd></dl> <p>Another equivalent (for graded lattices) condition is <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff</a>'s condition: </p> <dl><dd>for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> both cover <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\wedge y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\wedge y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f08a5192be1ce4d05be75c19a8d9ac7daa9a7777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.715ex; height:2.343ex;" alt="{\displaystyle x\wedge y,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\vee y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\vee y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304b00d1f1cf4a707c7863e8fae02a2dff7d5a8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle x\vee y}"></span> covers both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"></span></dd></dl> <p>A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span> exchanged, "covers" exchanged with "is covered by", and inequalities reversed.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Continuity_and_algebraicity">Continuity and algebraicity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=17" title="Edit section: Continuity and algebraicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of <a href="/wiki/Continuous_poset" title="Continuous poset">continuous posets</a>, consisting of posets where every element can be obtained as the supremum of a <a href="/wiki/Directed_set" title="Directed set">directed set</a> of elements that are <a href="/wiki/Way-below" class="mw-redirect" title="Way-below">way-below</a> the element. If one can additionally restrict these to the <a href="/wiki/Compact_element" title="Compact element">compact elements</a> of a poset for obtaining these directed sets, then the poset is even <a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">algebraic</a>. Both concepts can be applied to lattices as follows: </p> <ul><li>A <b><a href="/wiki/Continuous_lattice" class="mw-redirect" title="Continuous lattice">continuous lattice</a></b> is a complete lattice that is continuous as a poset.</li> <li>An <b><a href="/wiki/Algebraic_lattice" class="mw-redirect" title="Algebraic lattice">algebraic lattice</a></b> is a complete lattice that is algebraic as a poset.</li></ul> <p>Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via <a href="/wiki/Scott_information_system" title="Scott information system">Scott information systems</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Complements_and_pseudo-complements">Complements and pseudo-complements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=18" title="Edit section: Complements and pseudo-complements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Pseudocomplement" title="Pseudocomplement">pseudocomplement</a></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> be a bounded lattice with greatest element 1 and least element 0. Two elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> are <b>complements</b> of each other if and only if: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∨</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98f68444726069265e3216100b5f4c7e6342d3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.858ex; height:2.509ex;" alt="{\displaystyle x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.}"></span> </p><p>In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1/2,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1/2,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7cfb2fb275e418a9418c2b723d94a5765941582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.205ex; height:2.843ex;" alt="{\displaystyle \{0,1/2,1\}}"></span> with its usual ordering is a bounded lattice, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}"></span> does not have a complement. In the bounded lattice N<sub>5</sub>, the element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> has two complements, viz. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> (see Pic. 11). A bounded lattice for which every element has a complement is called a <a href="/wiki/Complemented_lattice" title="Complemented lattice">complemented lattice</a>. </p><p>A complemented lattice that is also distributive is a <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>. For a distributive lattice, the complement of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> when it exists, is unique. </p><p>In the case that the complement is unique, we write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lnot x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lnot x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcba0b38c73c5dcff4ecfdbd5a28f9254e27aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.134ex; height:2.009ex;" alt="{\textstyle \lnot x=y}"></span> and equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lnot y=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lnot y=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6761f658fe27e1824c48bb251496a89911652117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.781ex; height:2.009ex;" alt="{\textstyle \lnot y=x.}"></span> The corresponding unary <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"></span> called complementation, introduces an analogue of logical <a href="/wiki/Negation" title="Negation">negation</a> into lattice theory. </p><p><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebras</a> are an example of distributive lattices where some members might be lacking complements. Every element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> of a Heyting algebra has, on the other hand, a <a href="/wiki/Pseudo-complement" class="mw-redirect" title="Pseudo-complement">pseudo-complement</a>, also denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lnot x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lnot x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe8e66c59a33c1d54f1fb0cd607f7ffcada0b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.527ex; height:1.676ex;" alt="{\textstyle \lnot x.}"></span> The pseudo-complement is the greatest element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\wedge y=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∧</mo> <mi>y</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\wedge y=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c25e89f536819c102e105aa8f4a6aa9b309866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.976ex; height:2.509ex;" alt="{\displaystyle x\wedge y=0.}"></span> If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Jordan–Dedekind_chain_condition"><span id="Jordan.E2.80.93Dedekind_chain_condition"></span>Jordan–Dedekind chain condition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=19" title="Edit section: Jordan–Dedekind chain condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>chain</b> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span> is a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{x_{0},x_{1},\ldots ,x_{n}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{x_{0},x_{1},\ldots ,x_{n}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cba141a011d9453efee9b1ab5fecc47d87ca47c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.887ex; height:2.843ex;" alt="{\displaystyle \left\{x_{0},x_{1},\ldots ,x_{n}\right\},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>…</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05f6554dee997cb19abc3928ffbc13631894f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.464ex; height:2.176ex;" alt="{\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.}"></span> The <b>length</b> of this chain is <i>n</i>, or one less than its number of elements. A chain is <b>maximal</b> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> covers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db345bb67bd140474742faf5d2fff314daa04e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.23ex; height:2.009ex;" alt="{\displaystyle x_{i-1}}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e7f3cfcc75d64b89ecf7b8998713090bd9225e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.203ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n.}"></span> </p><p>If for any pair, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd9829c9ef4adcb0f9f5d53b27463a873a8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eecf73c55d04fd4e7005b8008f862185c93044a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.176ex;" alt="{\displaystyle x<y,}"></span> all maximal chains from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> have the same length, then the lattice is said to satisfy the <b>Jordan–Dedekind chain condition</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Graded/ranked"><span id="Graded.2Franked"></span>Graded/ranked</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=20" title="Edit section: Graded/ranked"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63147107331881c27e29102dfb49f3a745ccccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.234ex; height:2.843ex;" alt="{\displaystyle (L,\leq )}"></span> is called <b><a href="/wiki/Graded_poset" title="Graded poset">graded</a></b>, sometimes <b>ranked</b> (but see <a href="/wiki/Ranked_poset" title="Ranked poset">Ranked poset</a> for an alternative meaning), if it can be equipped with a <b>rank function</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:L\to \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>:</mo> <mi>L</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r:L\to \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e320b7b980e497e9020d37f051283f851f5dbbf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.861ex; height:2.176ex;" alt="{\displaystyle r:L\to \mathbb {N} }"></span> sometimes to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, compatible with the ordering (so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(x)<r(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(x)<r(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301fc2a6002846081737e59441528b1bea03cc24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.3ex; height:2.843ex;" alt="{\displaystyle r(x)<r(y)}"></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span>) such that whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> <a href="/wiki/Covering_relation" title="Covering relation">covers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(y)=r(x)+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(y)=r(x)+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdfb598b36be95e8186cef1931bc21f08473764e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.949ex; height:2.843ex;" alt="{\displaystyle r(y)=r(x)+1.}"></span> The value of the rank function for a lattice element is called its <b>rank</b>. </p><p>A lattice element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is said to <a href="/wiki/Covering_relation" title="Covering relation">cover</a> another element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb026bb3cb908d9dff974d2849d6c5181797e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.176ex;" alt="{\displaystyle y>x,}"></span> but there does not exist a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>z>x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mi>z</mi> <mo>></mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>z>x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c872e086e681ecfc0ec25b0f96677fa8f9f39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.417ex; height:2.176ex;" alt="{\displaystyle y>z>x.}"></span> Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22112f92b4b6d0c2f20283a6b5cb93e384091ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle y>x}"></span> means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637423b227b64dd6b186522e92ec84862b48e074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.23ex; height:2.676ex;" alt="{\displaystyle x\neq y.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Free_lattices">Free lattices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=21" title="Edit section: Free lattices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Free_lattice" title="Free lattice">Free lattice</a></div> <p>Any set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> may be used to generate the <b>free semilattice</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FX.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FX.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/481f67a1b62f0a1d0bcaff3bb699f0375b20fcab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.368ex; height:2.176ex;" alt="{\displaystyle FX.}"></span> The free semilattice is defined to consist of all of the finite subsets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"></span> with the semilattice operation given by ordinary <a href="/wiki/Set_union" class="mw-redirect" title="Set union">set union</a>. The free semilattice has the <a href="/wiki/Universal_property" title="Universal property">universal property</a>. For the <b>free lattice</b> over a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"></span> <a href="/wiki/Philip_M._Whitman" title="Philip M. Whitman">Whitman</a> gave a construction based on polynomials over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span><span class="nowrap" style="padding-left:0.1em;">'</span>s members.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Important_lattice-theoretic_notions">Important lattice-theoretic notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=22" title="Edit section: Important lattice-theoretic notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We now define some order-theoretic notions of importance to lattice theory. In the following, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> be an element of some lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6358b7808f1224473a219f51e5eede2495fa88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle L.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is called: </p> <ul><li><b>Join irreducible</b> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a\vee b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a\vee b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/517b6d3e4c55f185899a5058fe3cbb6e221f88c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.238ex; height:2.176ex;" alt="{\displaystyle x=a\vee b}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a{\text{ or }}x=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a{\text{ or }}x=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb64a69abc6e8dda38953ad136ec430da0cf9a7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.966ex; height:2.176ex;" alt="{\displaystyle x=a{\text{ or }}x=b.}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29aae3d806a3ec4ce45defc7365a525515971b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.332ex; height:2.509ex;" alt="{\displaystyle a,b\in L.}"></span> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> has a bottom element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95547343453ea34a314dd174f8458012f5a39ca3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 0,}"></span> some authors require <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span>.<sup id="cite_ref-FOOTNOTEDaveyPriestley200253_12-0" class="reference"><a href="#cite_note-FOOTNOTEDaveyPriestley200253-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> When the first condition is generalized to arbitrary joins <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{i\in I}a_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee _{i\in I}a_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3981a1ed11618db2f8efba99730583d4601f8e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.645ex; height:5.676ex;" alt="{\displaystyle \bigvee _{i\in I}a_{i},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is called <b>completely join irreducible</b> (or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b76220c6805c9b465d6efbc7686c624f49f3023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }"></span>-irreducible). The dual notion is <b>meet irreducibility</b> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caa4004cb216ef2930bb12fe805a76870caed94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }"></span>-irreducible). For example, in Pic. 2, the elements 2, 3, 4, and 5 are join irreducible, while 12, 15, 20, and 30 are meet irreducible. Depending on definition, the bottom element 1 and top element 60 may or may not be considered join irreducible and meet irreducible, respectively. In the lattice of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> with the usual order, each element is join irreducible, but none is completely join irreducible.</li> <li><b>Join prime</b> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq a\vee b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq a\vee b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9edb13a8bb8fb1da19dc870d6cef3acec28aa2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.238ex; height:2.343ex;" alt="{\displaystyle x\leq a\vee b}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq a{\text{ or }}x\leq b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq a{\text{ or }}x\leq b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a0c5004f0f89d503fdd197dcfc674616e94d73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.966ex; height:2.343ex;" alt="{\displaystyle x\leq a{\text{ or }}x\leq b.}"></span> Again some authors require <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span>, although this is unusual.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This too can be generalized to obtain the notion <b>completely join prime</b>. The dual notion is <b>meet prime</b>. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible. The converse holds if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is distributive.</li></ul> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> have a bottom element 0. An element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is an <a href="/wiki/Atom_(order_theory)" title="Atom (order theory)">atom</a> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b0ac4e2c02e608588f1525eac793c53eb7c55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle 0<x}"></span> and there exists no element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc54f30a0ec14db7f7f416a6f8f9ed651e6a676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.579ex; height:2.509ex;" alt="{\displaystyle y\in L}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<y<x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<y<x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70838217257cf695417498fdb9745dcb7d6e6649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.491ex; height:2.509ex;" alt="{\displaystyle 0<y<x.}"></span> Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is called: </p> <ul><li><a href="/wiki/Atomic_(order_theory)" class="mw-redirect" title="Atomic (order theory)">Atomic</a> if for every nonzero element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efcf5b47104c2ddf893d7608183b766a281cea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.23ex; height:2.509ex;" alt="{\displaystyle L,}"></span> there exists an atom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq x;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq x;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896b4e177a4b050084dd6658f49dfea54180f9ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.305ex; height:2.343ex;" alt="{\displaystyle a\leq x;}"></span><sup id="cite_ref-FOOTNOTEGrätzer2003246Exercise_3_14-0" class="reference"><a href="#cite_note-FOOTNOTEGrätzer2003246Exercise_3-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Atomistic_(order_theory)" class="mw-redirect" title="Atomistic (order theory)">Atomistic</a> if every element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> of atoms.<sup id="cite_ref-FOOTNOTEGrätzer2003234after_Def.1_15-0" class="reference"><a href="#cite_note-FOOTNOTEGrätzer2003234after_Def.1-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li></ul> <p>However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Give an example (September 2022)">citation needed</span></a></i>]</sup> </p><p>The notions of <a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideals</a> and the dual notion of <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filters</a> refer to particular kinds of <a href="/wiki/Subset" title="Subset">subsets</a> of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a> – Concept in order theory</li> <li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a> – Concept in mathematics</li> <li><a href="/wiki/Orthocomplemented_lattice" class="mw-redirect" title="Orthocomplemented lattice">Orthocomplemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a> – Order whose elements are all comparable</li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a> – Nonempty, upper-bounded, downward-closed subset and <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filter</a> (dual notions)</li> <li><a href="/wiki/Skew_lattice" title="Skew lattice">Skew lattice</a> – Algebraic Structure<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span> (generalization to non-commutative join and meet)</li> <li><a href="/wiki/Eulerian_lattice" class="mw-redirect" title="Eulerian lattice">Eulerian lattice</a></li> <li><a href="/wiki/Post%27s_lattice" title="Post's lattice">Post's lattice</a> – lattice of all clones (sets of logical connectives closed under composition and containing all projections) on a two-element set {0, 1}, ordered by inclusion<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Tamari_lattice" title="Tamari lattice">Tamari lattice</a> – mathematical object formed by an order on the way of parenthesing an expression<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Young%E2%80%93Fibonacci_lattice" title="Young–Fibonacci lattice">Young–Fibonacci lattice</a></li> <li><a href="/wiki/0,1-simple_lattice" title="0,1-simple lattice">0,1-simple lattice</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Applications_that_use_lattice_theory">Applications that use lattice theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=24" title="Edit section: Applications that use lattice theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Prose plainlinks metadata ambox ambox-style ambox-Prose" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>is in <a href="/wiki/MOS:LIST" class="mw-redirect" title="MOS:LIST">list</a> format but may read better as <a href="/wiki/MOS:PROSE" class="mw-redirect" title="MOS:PROSE">prose</a></b>.<span class="hide-when-compact"> You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Lattice_(order)&action=edit">converting this article</a>, if appropriate. <a href="/wiki/Help:Editing" title="Help:Editing">Editing help</a> is available.</span> <span class="date-container"><i>(<span class="date">March 2017</span>)</i></span></div></td></tr></tbody></table> <p><i>Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.</i> </p> <ul><li><a href="/wiki/Pointless_topology" title="Pointless topology">Pointless topology</a></li> <li><a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">Lattice of subgroups</a></li> <li><a href="/wiki/Spectral_space" title="Spectral space">Spectral space</a></li> <li><a href="/wiki/Invariant_subspace" title="Invariant subspace">Invariant subspace</a></li> <li><a href="/wiki/Closure_operator" title="Closure operator">Closure operator</a></li> <li><a href="/wiki/Abstract_interpretation" title="Abstract interpretation">Abstract interpretation</a></li> <li><a href="/wiki/Subsumption_lattice" title="Subsumption lattice">Subsumption lattice</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy set</a> theory</li> <li><a href="/wiki/First-order_logic#Algebraizations" title="First-order logic">Algebraizations of first-order logic</a></li> <li><a href="/wiki/Semantics_of_programming_languages" class="mw-redirect" title="Semantics of programming languages">Semantics of programming languages</a></li> <li><a href="/wiki/Domain_theory" title="Domain theory">Domain theory</a></li> <li><a href="/wiki/Ontology_(computer_science)" class="mw-redirect" title="Ontology (computer science)">Ontology (computer science)</a></li> <li><a href="/wiki/Multiple_inheritance" title="Multiple inheritance">Multiple inheritance</a></li> <li><a href="/wiki/Formal_concept_analysis" title="Formal concept analysis">Formal concept analysis</a> and <a href="/wiki/Lattice_Miner" title="Lattice Miner">Lattice Miner</a> (theory and tool)</li> <li><a href="/wiki/Bloom_filter#Compact_approximators" title="Bloom filter">Bloom filter</a></li> <li><a href="/wiki/Information_flow" title="Information flow">Information flow</a></li> <li><a href="/wiki/Ordinal_optimization" class="mw-redirect" title="Ordinal optimization">Ordinal optimization</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Median_graph" title="Median graph">Median graph</a></li> <li><a href="/wiki/Knowledge_space" title="Knowledge space">Knowledge space</a></li> <li><a href="/wiki/Induction_of_regular_languages#Lattice_of_automata" title="Induction of regular languages">Regular language learning</a></li> <li><a href="/wiki/Analogical_modeling" title="Analogical modeling">Analogical modeling</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=25" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEGrätzer2003[httpsbooksgooglecombooksidSoGLVCPuOz0CpgPA52_52]-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrätzer2003[httpsbooksgooglecombooksidSoGLVCPuOz0CpgPA52_52]_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrätzer2003">Grätzer 2003</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SoGLVCPuOz0C&pg=PA52">52</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFBirkhoff1948">Birkhoff 1948</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.166886/page/n35/mode/2up">18</a>. "since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=a\vee (a\wedge (a\vee a))=a\vee a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∨</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>∨</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a\vee (a\wedge (a\vee a))=a\vee a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ab2f6e6ab4be5c1b7a7a45566916b867d7e4075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.755ex; height:2.843ex;" alt="{\displaystyle a=a\vee (a\wedge (a\vee a))=a\vee a}"></span> and dually". Birkhoff attributes this to <a href="#CITEREFDedekind1897">Dedekind 1897</a>, p. <a rel="nofollow" class="external text" href="https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf#page=10">8</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Burris, Stanley N., and Sankappanavar, H. P., 1981. <a rel="nofollow" class="external text" href="http://www.thoralf.uwaterloo.ca/htdocs/ualg.html"><i>A Course in Universal Algebra</i>.</a> Springer-Verlag. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-90578-2" title="Special:BookSources/3-540-90578-2">3-540-90578-2</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaker2010" class="citation web cs1">Baker, Kirby (2010). <a rel="nofollow" class="external text" href="https://www.math.ucla.edu/~baker/222a/handouts/s_complete.pdf">"Complete Lattices"</a> <span class="cs1-format">(PDF)</span>. <i>UCLA Department of Mathematics</i><span class="reference-accessdate">. Retrieved <span class="nowrap">8 June</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=UCLA+Department+of+Mathematics&rft.atitle=Complete+Lattices&rft.date=2010&rft.aulast=Baker&rft.aufirst=Kirby&rft_id=https%3A%2F%2Fwww.math.ucla.edu%2F~baker%2F222a%2Fhandouts%2Fs_complete.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplansky1972" class="citation book cs1">Kaplansky, Irving (1972). <i>Set Theory and Metric Spaces</i> (2nd ed.). New York City: <a href="/wiki/Chelsea_Publishing_Company" title="Chelsea Publishing Company">AMS Chelsea Publishing</a>. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821826942" title="Special:BookSources/9780821826942"><bdi>9780821826942</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Metric+Spaces&rft.place=New+York+City&rft.pages=14&rft.edition=2nd&rft.pub=AMS+Chelsea+Publishing&rft.date=1972&rft.isbn=9780821826942&rft.aulast=Kaplansky&rft.aufirst=Irving&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFDaveyPriestley2002">Davey & Priestley (2002)</a>, Exercise 4.1, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA104">p. 104</a>.</span> </li> <li id="cite_note-Davey.Priestley.2002.10.6-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Davey.Priestley.2002.10.6_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Davey.Priestley.2002.10.6_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDaveyPriestley2002">Davey & Priestley (2002)</a>, Theorem 4.10, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA89">p. 89</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFDaveyPriestley2002">Davey & Priestley (2002)</a>, Theorem 10.21, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA238">pp. 238–239</a>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley1997" class="citation cs2"><a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">Stanley, Richard P</a> (1997), <i>Enumerative Combinatorics (vol. 1)</i>, Cambridge University Press, pp. 103–104, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66351-2" title="Special:BookSources/0-521-66351-2"><bdi>0-521-66351-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+Combinatorics+%28vol.+1%29&rft.pages=103-104&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-66351-2&rft.aulast=Stanley&rft.aufirst=Richard+P&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhilip_Whitman1941" class="citation journal cs1">Philip Whitman (1941). "Free Lattices I". <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. <b>42</b> (1): 325–329. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969001">10.2307/1969001</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969001">1969001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Free+Lattices+I&rft.volume=42&rft.issue=1&rft.pages=325-329&rft.date=1941&rft_id=info%3Adoi%2F10.2307%2F1969001&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969001%23id-name%3DJSTOR&rft.au=Philip+Whitman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhilip_Whitman1942" class="citation journal cs1">Philip Whitman (1942). "Free Lattices II". <i>Annals of Mathematics</i>. <b>43</b> (1): 104–115. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968883">10.2307/1968883</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968883">1968883</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Free+Lattices+II&rft.volume=43&rft.issue=1&rft.pages=104-115&rft.date=1942&rft_id=info%3Adoi%2F10.2307%2F1968883&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968883%23id-name%3DJSTOR&rft.au=Philip+Whitman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEDaveyPriestley200253-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDaveyPriestley200253_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDaveyPriestley2002">Davey & Priestley 2002</a>, p. 53.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoffmann1981" class="citation conference cs1">Hoffmann, Rudolf-E. (1981). <i>Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications</i>. Continuous Lattices. Vol. 871. pp. 159–208. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0089907">10.1007/BFb0089907</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Continuous+posets%2C+prime+spectra+of+completely+distributive+complete+lattices%2C+and+Hausdorff+compactifications&rft.pages=159-208&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBFb0089907&rft.aulast=Hoffmann&rft.aufirst=Rudolf-E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGrätzer2003246Exercise_3-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrätzer2003246Exercise_3_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrätzer2003">Grätzer 2003</a>, p. 246, Exercise 3.</span> </li> <li id="cite_note-FOOTNOTEGrätzer2003234after_Def.1-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrätzer2003234after_Def.1_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrätzer2003">Grätzer 2003</a>, p. 234, after Def.1.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <p>Monographs available free online: </p> <ul><li>Burris, Stanley N., and Sankappanavar, H. P., 1981. <i><a rel="nofollow" class="external text" href="http://www.thoralf.uwaterloo.ca/htdocs/ualg.html">A Course in Universal Algebra.</a></i> Springer-Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-90578-2" title="Special:BookSources/3-540-90578-2">3-540-90578-2</a>.</li> <li>Jipsen, Peter, and Henry Rose, <i><a rel="nofollow" class="external text" href="http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html">Varieties of Lattices</a></i>, Lecture Notes in Mathematics 1533, Springer Verlag, 1992. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-56314-8" title="Special:BookSources/0-387-56314-8">0-387-56314-8</a>.</li></ul> <p>Elementary texts recommended for those with limited <a href="/wiki/Mathematical_maturity" title="Mathematical maturity">mathematical maturity</a>: </p> <ul><li>Donnellan, Thomas, 1968. <i>Lattice Theory</i>. Pergamon.</li> <li><a href="/wiki/George_Gr%C3%A4tzer" title="George Grätzer">Grätzer, George</a>, 1971. <i>Lattice Theory: First concepts and distributive lattices</i>. W. H. Freeman.</li></ul> <p>The standard contemporary introductory text, somewhat harder than the above: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaveyPriestley2002" class="citation cs2">Davey, B. A.; <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">Priestley, H. A.</a> (2002), <a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order"><i>Introduction to Lattices and Order</i></a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78451-1" title="Special:BookSources/978-0-521-78451-1"><bdi>978-0-521-78451-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Lattices+and+Order&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-78451-1&rft.aulast=Davey&rft.aufirst=B.+A.&rft.au=Priestley%2C+H.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></li></ul> <p>Advanced monographs: </p> <ul><li><a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>, 1967. <i>Lattice Theory</i>, 3rd ed. Vol. 25 of AMS Colloquium Publications. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>.</li> <li><a href="/wiki/Robert_P._Dilworth" title="Robert P. Dilworth">Robert P. Dilworth</a> and Crawley, Peter, 1973. <i>Algebraic Theory of Lattices</i>. Prentice-Hall. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-022269-5" title="Special:BookSources/978-0-13-022269-5">978-0-13-022269-5</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrätzer2003" class="citation book cs1">Grätzer, George (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/generallatticeth0000grat"><i>General Lattice Theory</i></a></span> (Second ed.). Basel: Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-6996-5" title="Special:BookSources/978-3-7643-6996-5"><bdi>978-3-7643-6996-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Lattice+Theory&rft.place=Basel&rft.edition=Second&rft.pub=Birkh%C3%A4user&rft.date=2003&rft.isbn=978-3-7643-6996-5&rft.aulast=Gr%C3%A4tzer&rft.aufirst=George&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgenerallatticeth0000grat&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></li></ul> <p>On free lattices: </p> <ul><li>R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>.</li> <li><a href="/wiki/Peter_Johnstone_(mathematician)" title="Peter Johnstone (mathematician)">Johnstone, P. T.</a>, 1982. <i>Stone spaces</i>. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.</li></ul> <p>On the history of lattice theory: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŠtĕpánka_Bilová2001" class="citation book cs1">Štĕpánka Bilová (2001). Eduard Fuchs (ed.). <a rel="nofollow" class="external text" href="http://dml.cz/bitstream/handle/10338.dmlcz/401261/DejinyMat_17-2001-1_31.pdf"><i>Lattice theory — its birth and life</i></a> <span class="cs1-format">(PDF)</span>. Prometheus. pp. 250–257.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lattice+theory+%E2%80%94+its+birth+and+life&rft.pages=250-257&rft.pub=Prometheus&rft.date=2001&rft.au=%C5%A0t%C4%95p%C3%A1nka+Bilov%C3%A1&rft_id=http%3A%2F%2Fdml.cz%2Fbitstream%2Fhandle%2F10338.dmlcz%2F401261%2FDejinyMat_17-2001-1_31.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirkhoff1948" class="citation book cs1">Birkhoff, Garrett (1948). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.166886/"><i>Lattice Theory</i></a> (2nd ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lattice+Theory&rft.edition=2nd&rft.date=1948&rft.aulast=Birkhoff&rft.aufirst=Garrett&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.166886%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span> Textbook with numerous attributions in the footnotes.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlimm2011" class="citation journal cs1">Schlimm, Dirk (November 2011). "On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others". <i>Synthese</i>. <b>183</b> (1): 47–68. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.594.8898">10.1.1.594.8898</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11229-009-9667-9">10.1007/s11229-009-9667-9</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11012081">11012081</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Synthese&rft.atitle=On+the+creative+role+of+axiomatics.+The+discovery+of+lattices+by+Schr%C3%B6der%2C+Dedekind%2C+Birkhoff%2C+and+others&rft.volume=183&rft.issue=1&rft.pages=47-68&rft.date=2011-11&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.594.8898%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11012081%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11229-009-9667-9&rft.aulast=Schlimm&rft.aufirst=Dirk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span> Summary of the history of lattices.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekind1897" class="citation cs2"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a> (1897), <a rel="nofollow" class="external text" href="https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf">"Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler"</a> <span class="cs1-format">(PDF)</span>, <i>Braunschweiger Festschrift</i>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24355%2Fdbbs.084-200908140200-2">10.24355/dbbs.084-200908140200-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Braunschweiger+Festschrift&rft.atitle=%C3%9Cber+Zerlegungen+von+Zahlen+durch+ihre+gr%C3%B6ssten+gemeinsamen+Teiler&rft.date=1897&rft_id=info%3Adoi%2F10.24355%2Fdbbs.084-200908140200-2&rft.aulast=Dedekind&rft.aufirst=Richard&rft_id=https%3A%2F%2Fpublikationsserver.tu-braunschweig.de%2Fservlets%2FMCRFileNodeServlet%2Fdbbs_derivate_00006737%2FV.C.1596.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></li></ul> <p>On applications of lattice theory: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarrett_Birkhoff1967" class="citation book cs1">Garrett Birkhoff (1967). James C. Abbot (ed.). <i>What can Lattices do for you?</i>. Van Nostrand.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=What+can+Lattices+do+for+you%3F&rft.pub=Van+Nostrand&rft.date=1967&rft.au=Garrett+Birkhoff&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170808093034/http://tocs.ulb.tu-darmstadt.de/129983330.pdf">Table of contents</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lattice_(order)&action=edit&section=27" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Lattice-ordered_group">"Lattice-ordered group"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lattice-ordered+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLattice-ordered_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Lattice"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Lattice.html">"Lattice"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Lattice&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLattice.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALattice+%28order%29" class="Z3988"></span></span></li> <li>J.B. Nation, <a rel="nofollow" class="external text" href="https://math.hawaii.edu/~jb/"><i>Notes on Lattice Theory</i></a>, course notes, revised 2017.</li> <li>Ralph Freese, <a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/LatThy/">"Lattice Theory Homepage"</a>.</li> <li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A006966">sequence A006966 (Number of unlabeled lattices with <i>n</i> elements)</a></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Order_theory" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order_theory" title="Template:Order theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a class="mw-selflink selflink">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a class="mw-selflink selflink">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div> <div 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