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#aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Appendix:Glossary_of_order_theory" class="extiw" title="wiktionary:Appendix:Glossary of order theory">Appendix:Glossary of order theory</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <p>This is a glossary of some terms used in various branches of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that are related to the fields of <a href="/wiki/Order_theory" title="Order theory">order</a>, <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a>, and <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>. Note that there is a structured <a href="/wiki/List_of_order_topics" class="mw-redirect" title="List of order topics">list of order topics</a> available as well. Other helpful resources might be the following overview articles: </p> <ul><li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completeness properties</a> of partial orders</li> <li><a href="/wiki/Distributivity_(order_theory)" title="Distributivity (order theory)">distributivity laws</a> of order theory</li> <li><a href="/wiki/Limit_preserving_(order_theory)" class="mw-redirect" title="Limit preserving (order theory)">preservation properties</a> of functions between posets.</li></ul> <p>In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, <span class="mwe-math-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> <mspace width="thinmathspace" /> </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/24112548985eab096493f73f838580442780b57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,\leq \,}"></span> will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the <a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">strict order</a> induced 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/0f781d78c3eceb2d86f2b016d07c794adaa447f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.842ex; height:2.176ex;" alt="{\displaystyle \,\leq .}"></span> </p> <div class="noprint"><div role="navigation" id="toc" class="toc plainlinks" aria-labelledby="tocheading" style="text-align:left;"><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 .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"> <div id="toctitle" class="toctitle" style="text-align:center;display:inline-block;"><span id="tocheading" style="font-weight:bold;">Contents: </span></div> <div style="margin:auto; display:inline-block;"> <ul><li><a href="#top">Top</a></li> <li><a href="#0–9">0–9</a></li> <li><a href="#A">A</a></li> <li><a href="#B">B</a></li> <li><a href="#C">C</a></li> <li><a href="#D">D</a></li> <li><a href="#E">E</a></li> <li><a href="#F">F</a></li> <li><a href="#G">G</a></li> <li><a href="#H">H</a></li> <li><a href="#I">I</a></li> <li><a href="#J">J</a></li> <li><a href="#K">K</a></li> <li><a href="#L">L</a></li> <li><a href="#M">M</a></li> <li><a href="#N">N</a></li> <li><a href="#O">O</a></li> <li><a href="#P">P</a></li> <li><a href="#Q">Q</a></li> <li><a href="#R">R</a></li> <li><a href="#S">S</a></li> <li><a href="#T">T</a></li> <li><a href="#U">U</a></li> <li><a href="#V">V</a></li> <li><a href="#W">W</a></li> <li><a href="#X">X</a></li> <li><a href="#Y">Y</a></li> <li><a href="#Z">Z</a> </li></ul> <p class="mw-empty-elt"> </p> </div></div></div></div> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="A">A</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=1" title="Edit section: A"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Acyclic</b>. A <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> is acyclic if it contains no "cycles": equivalently, its <a href="/wiki/Transitive_closure" title="Transitive closure">transitive closure</a> is <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>.<sup id="cite_ref-BosSuz_1-0" class="reference"><a href="#cite_note-BosSuz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><b>Adjoint</b>. See <i>Galois connection</i>.</li> <li><b><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a></b>. For a preordered set <i>P</i>, any upper set <i>O</i> is <b>Alexandrov-open</b>. Inversely, a topology is Alexandrov if any intersection of open sets is open.</li> <li><b><a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">Algebraic poset</a></b>. A poset is algebraic if it has a base of compact elements.</li> <li><b><a href="/wiki/Antichain" title="Antichain">Antichain</a></b>. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements <i>x</i> and <i>y</i> such that <i>x</i> ≤ <i>y</i>. In other words, the order relation of an antichain is just the identity relation.</li> <li><b>Approximates relation</b>. See <i>way-below relation</i>.</li> <li><b>Antisymmetric relation</b>. A <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> <i>R</i> on a set <i>X</i> is <b><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a></b>, if <i>x R y</i> and <i>y R x</i> implies <i>x = y</i>, for all elements <i>x</i>, <i>y</i> in <i>X</i>.</li> <li><b>Antitone</b>. An <a href="/wiki/Antitone" class="mw-redirect" title="Antitone">antitone</a> function <i>f</i> between posets <i>P</i> and <i>Q</i> is a function for which, for all elements <i>x</i>, <i>y</i> of <i>P</i>, <i>x</i> ≤ <i>y</i> (in <i>P</i>) implies <i>f</i>(<i>y</i>) ≤ <i>f</i>(<i>x</i>) (in <i>Q</i>). Another name for this property is <i>order-reversing</i>. In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, in the presence of <a href="/wiki/Total_order" title="Total order">total orders</a>, such functions are often called <b>monotonically decreasing</b>, but this is not a very convenient description when dealing with non-total orders. The dual notion is called <i>monotone</i> or <i>order-preserving</i>.</li> <li><b><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric relation</a></b>. A <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> <i>R</i> on a set <i>X</i> is asymmetric, if <i>x R y</i> implies <i>not y R x</i>, for all elements <i>x</i>, <i>y</i> in <i>X</i>.</li> <li><b><a href="/wiki/Atom_(order_theory)" title="Atom (order theory)">Atom</a></b>. An atom in a poset <i>P</i> with least element 0, is an element that is minimal among all elements that are unequal to 0.</li> <li><b>Atomic</b>. An atomic poset <i>P</i> with least element 0 is one in which, for every non-zero element <i>x</i> of <i>P</i>, there is an atom <i>a</i> of <i>P</i> with <i>a</i> ≤ <i>x</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="B">B</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=2" title="Edit section: B"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Base</b>. See <i>continuous poset</i>.</li> <li><b><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></b>. A binary relation over two sets <span class="mwe-math-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{\text{ and }}Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X{\text{ and }}Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d30e859e9af06fa0c05b5a7ba025f86f6f24f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.662ex; height:2.176ex;" alt="{\displaystyle X{\text{ and }}Y}"></span> is a subset of their <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</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\times 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\times Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f39ef78be28dc2b9015ff7f82e9a1ef719a9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.241ex; height:2.176ex;" alt="{\displaystyle X\times Y.}"></span></li> <li><b><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></b>. A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element <i>x</i> has a complement ¬<i>x</i>, such that <i>x</i> ∧ ¬<i>x</i> = 0 and <i>x</i> ∨ ¬<i>x</i> = 1.</li> <li><b><a href="/wiki/Bounded_poset" class="mw-redirect" title="Bounded poset">Bounded poset</a></b>. A <a href="/wiki/Bounded_poset" class="mw-redirect" title="Bounded poset">bounded</a> poset is one that has a least element and a greatest element.</li> <li><b><a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">Bounded complete</a></b>. A poset is <a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">bounded complete</a> if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.</li></ul> <div class="mw-heading mw-heading2"><h2 id="C">C</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=3" title="Edit section: C"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Total_order#Chains" title="Total order">Chain</a></b>. A chain is a totally ordered set or a totally ordered subset of a poset. See also <em>total order</em>.</li> <li><b><a href="/wiki/Chain_complete" class="mw-redirect" title="Chain complete">Chain complete</a></b>. A <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> in which every <a href="/wiki/Total_order#Chains" title="Total order">chain</a> has a <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a>.</li> <li><b><a href="/wiki/Closure_operator" title="Closure operator">Closure operator</a></b>. A closure operator on the poset <i>P</i> is a function <i>C</i> : <i>P</i> → <i>P</i> that is monotone, <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>, and satisfies <i>C</i>(<i>x</i>) ≥ <i>x</i> for all <i>x</i> in <i>P</i>.</li> <li><b><a href="/wiki/Compact_element" title="Compact element">Compact</a></b>. An element <i>x</i> of a poset is compact if it is <i><a href="/wiki/Way-below_relation" class="mw-redirect" title="Way-below relation">way below</a></i> itself, i.e. <i>x</i><<<i>x</i>. One also says that such an <i>x</i> is <em>finite</em>.</li> <li><b><a href="/wiki/Comparability" title="Comparability">Comparable</a></b>. Two elements <i>x</i> and <i>y</i> of a poset <i>P</i> are comparable if either <i>x</i> ≤ <i>y</i> or <i>y</i> ≤ <i>x</i>.</li> <li><b><a href="/wiki/Comparability_graph" title="Comparability graph">Comparability graph</a></b>. The comparability graph of a poset (<i>P</i>, ≤) is the <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> with vertex set <i>P</i> in which the edges are those pairs of distinct elements of <i>P</i> that are comparable under ≤ (and, in particular, under its reflexive reduction <).</li> <li><b><a href="/wiki/Complete_Boolean_algebra" title="Complete Boolean algebra">Complete Boolean algebra</a></b>. A <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> that is a complete lattice.</li> <li><b><a href="/wiki/Complete_Heyting_algebra" title="Complete Heyting algebra">Complete Heyting algebra</a></b>. A <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a> that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts <i>frame</i> and <i>locale</i>.</li> <li><b><a href="/wiki/Complete_lattice" title="Complete lattice">Complete lattice</a></b>. A complete <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.</li> <li><b><a href="/wiki/Complete_partial_order" title="Complete partial order">Complete partial order</a></b>. A complete partial order, or <b>cpo</b>, is a <a href="/wiki/Directed_complete_partial_order" class="mw-redirect" title="Directed complete partial order">directed complete partial order</a> (q.v.) with least element.</li> <li><b>Complete relation</b>. Synonym for <i><a href="/wiki/Connected_relation" title="Connected relation">Connected relation</a></i>.</li> <li><b>Complete semilattice</b>. The notion of a <i>complete semilattice</i> is defined in different ways. As explained in the article on <a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completeness (order theory)</a>, any poset for which either all suprema or all infima exist is already a complete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of a complete lattice. In other cases, complete (meet-) semilattices are defined to be <a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">bounded complete</a> <a href="/wiki/Complete_partial_order" title="Complete partial order">cpos</a>, which is arguably the most complete class of posets that are not already complete lattices.</li> <li><b><a href="/wiki/Completely_distributive_lattice" title="Completely distributive lattice">Completely distributive lattice</a></b>. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.</li> <li><b>Completion</b>. A completion of a poset is an <a href="/wiki/Order-embedding" class="mw-redirect" title="Order-embedding">order-embedding</a> of the poset in a complete lattice.</li> <li><b><a href="/wiki/Dedekind%E2%80%93MacNeille_completion" title="Dedekind–MacNeille completion">Completion by cuts</a></b>. Synonym of <a href="/wiki/Dedekind%E2%80%93MacNeille_completion" title="Dedekind–MacNeille completion">Dedekind–MacNeille completion</a>.</li> <li><b><a href="/wiki/Connected_relation" title="Connected relation">Connected relation</a></b>. A total or complete relation <i>R</i> on a set <i>X</i> has the property that for all elements <i>x</i>, <i>y</i> of <i>X</i>, at least one of <i>x R y</i> or <i>y R x</i> holds.</li> <li><b><a href="/wiki/Continuous_poset" title="Continuous poset">Continuous poset</a></b>. A poset is continuous if it has a <b>base</b>, i.e. a subset <i>B</i> of <i>P</i> such that every element <i>x</i> of <i>P</i> is the supremum of a directed set contained in {<i>y</i> in <i>B</i> | <i>y</i><<<i>x</i>}.</li> <li><b>Continuous function</b>. See <i>Scott-continuous</i>.</li> <li><b>Converse</b>. The converse <° of an order < is that in which x <° y whenever y < x.</li> <li><b>Cover</b>. An element <i>y</i> of a poset <i>P</i> is said to cover an element <i>x</i> of <i>P</i> (and is called a cover of <i>x</i>) if <i>x</i> < <i>y</i> and there is no element <i>z</i> of <i>P</i> such that <i>x</i> < <i>z</i> < <i>y</i>.</li> <li><b><a href="/wiki/Complete_partial_order" title="Complete partial order">cpo</a></b>. See <i>complete partial order</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="D">D</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=4" title="Edit section: D"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>dcpo</b>. See <i><a href="/wiki/Directed_complete_partial_order" class="mw-redirect" title="Directed complete partial order">directed complete partial order</a></i>.</li> <li><b><a href="/wiki/Dedekind%E2%80%93MacNeille_completion" title="Dedekind–MacNeille completion">Dedekind–MacNeille completion</a></b>. The Dedekind–MacNeille completion of a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> is the smallest <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> that contains it.</li> <li><b><a href="/wiki/Dense_order" title="Dense order">Dense order</a></b>. A <b><a href="/wiki/Dense_order" title="Dense order">dense</a></b> poset <i>P</i> is one in which, for all elements <i>x</i> and <i>y</i> in <i>P</i> with <i>x</i> < <i>y</i>, there is an element <i>z</i> in <i>P</i>, such that <i>x</i> < <i>z</i> < <i>y</i>. A subset <i>Q</i> of <i>P</i> is <b>dense in</b> <i>P</i> if for any elements <i>x</i> < <i>y</i> in <i>P</i>, there is an element <i>z</i> in <i>Q</i> such that <i>x</i> < <i>z</i> < <i>y</i>.</li> <li><b><a href="/wiki/Derangement" title="Derangement">Derangement</a></b>. A permutation of the elements of a set, such that no element appears in its original position.</li> <li><b><a href="/wiki/Directed_set" title="Directed set">Directed set</a></b>. A <a href="/wiki/Non-empty" class="mw-redirect" title="Non-empty">non-empty</a> subset <i>X</i> of a poset <i>P</i> is called directed, if, for all elements <i>x</i> and <i>y</i> of <i>X</i>, there is an element <i>z</i> of <i>X</i> such that <i>x</i> ≤ <i>z</i> and <i>y</i> ≤ <i>z</i>. The dual notion is called <i>filtered</i>.</li> <li><b><a href="/wiki/Directed_complete_partial_order" class="mw-redirect" title="Directed complete partial order">Directed complete partial order</a></b>. A poset <i>D</i> is said to be a directed complete poset, or <b>dcpo</b>, if every directed subset of <i>D</i> has a supremum.</li> <li><b><a href="/wiki/Distributivity_(order_theory)" title="Distributivity (order theory)">Distributive</a></b>. A lattice <i>L</i> is called distributive if, for all <i>x</i>, <i>y</i>, and <i>z</i> in <i>L</i>, we find that <i>x</i> ∧ (<i>y</i> ∨ <i>z</i>) = (<i>x</i> ∧ <i>y</i>) ∨ (<i>x</i> ∧ <i>z</i>). This condition is known to be equivalent to its order dual. A meet-<a href="/wiki/Semilattice" title="Semilattice">semilattice</a> is distributive if for all elements <i>a</i>, <i>b</i> and <i>x</i>, <i>a</i> ∧ <i>b</i> ≤ <i>x</i> implies the existence of elements <i>a' </i> ≥ <i>a</i> and <i>b' </i> ≥ <i>b</i> such that <i>a' </i> ∧ <i>b' </i> = <i>x</i>. See also <i>completely distributive</i>.</li> <li><b><a href="/wiki/Domain_theory" title="Domain theory">Domain</a></b>. Domain is a general term for objects like those that are studied in <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>. If used, it requires further definition.</li> <li><b>Down-set</b>. See <i>lower set</i>.</li> <li><b><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Dual</a></b>. For a poset (<i>P</i>, ≤), the dual order <i>P</i><sup><i>d</i></sup> = (<i>P</i>, ≥) is defined by setting <i>x ≥ y</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>y ≤ x</i>. The dual order of <i>P</i> is sometimes denoted by <i>P</i><sup>op</sup>, and is also called <i>opposite</i> or <i>converse</i> order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual of a given set. This exchanges ≤ and ≥, meets and joins, zero and unit.</li></ul> <div class="mw-heading mw-heading2"><h2 id="E">E</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=5" title="Edit section: E"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Extension</b>. For partial orders ≤ and ≤′ on a set <i>X</i>, ≤′ is an extension of ≤ provided that for all elements <i>x</i> and <i>y</i> of <i>X</i>, <i>x</i> ≤ <i>y</i> implies that <i>x</i> ≤′ <i>y</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="F">F</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=6" title="Edit section: F"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></b>. A subset <i>X</i> of a poset <i>P</i> is called a filter if it is a filtered upper set. The dual notion is called <i>ideal</i>.</li> <li><b>Filtered</b>. A <a href="/wiki/Non-empty" class="mw-redirect" title="Non-empty">non-empty</a> subset <i>X</i> of a poset <i>P</i> is called filtered, if, for all elements <i>x</i> and <i>y</i> of <i>X</i>, there is an element <i>z</i> of <i>X</i> such that <i>z</i> ≤ <i>x</i> and <i>z</i> ≤ <i>y</i>. The dual notion is called <i>directed</i>.</li> <li><b>Finite element</b>. See <i>compact</i>.</li> <li><b><a href="/wiki/Complete_Heyting_algebra" title="Complete Heyting algebra">Frame</a></b>. A frame <i>F</i> is a complete lattice, in which, for every <i>x</i> in <i>F</i> and every subset <i>Y</i> of <i>F</i>, the infinite distributive law <i>x</i> ∧ <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424eb787b9be7b652deb858148ac5412c317aebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigvee }"></span><i>Y</i> = <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424eb787b9be7b652deb858148ac5412c317aebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigvee }"></span>{<i>x</i> ∧ <i>y</i> | <i>y</i> in <i>Y</i>} holds. Frames are also known as <i>locales</i> and as complete <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebras</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="G">G</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=7" title="Edit section: G"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a></b>. Given two posets <i>P</i> and <i>Q</i>, a pair of monotone functions <i>F</i>:<i>P</i> → <i>Q</i> and <i>G</i>:<i>Q</i> → <i>P</i> is called a Galois connection, if <i>F</i>(<i>x</i>) ≤ <i>y</i> is equivalent to <i>x</i> ≤ <i>G</i>(<i>y</i>), for all <i>x</i> in <i>P</i> and <i>y</i> in <i>Q</i>. <i>F</i> is called the <b>lower adjoint</b> of <i>G</i> and <i>G</i> is called the <b>upper adjoint</b> of <i>F</i>.</li> <li><b><a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">Greatest element</a></b>. For a subset <i>X</i> of a poset <i>P</i>, an element <i>a</i> of <i>X</i> is called the greatest element of <i>X</i>, if <i>x</i> ≤ <i>a</i> for every element <i>x</i> in <i>X</i>. The dual notion is called <i>least element</i>.</li> <li><b>Ground set</b>. The ground set of a poset (<i>X</i>, ≤) is the set <i>X</i> on which the partial order ≤ is defined.</li></ul> <div class="mw-heading mw-heading2"><h2 id="H">H</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=8" title="Edit section: H"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></b>. A Heyting algebra <i>H</i> is a bounded lattice in which the function <i>f</i><sub><i>a</i></sub>: <i>H</i> → <i>H</i>, given by <i>f</i><sub><i>a</i></sub>(<i>x</i>) = <i>a</i> ∧ <i>x</i> is the lower adjoint of a <a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a>, for every element <i>a</i> of <i>H</i>. The upper adjoint of <i>f</i><sub><i>a</i></sub> is then denoted by <i>g</i><sub><i>a</i></sub>, with <i>g</i><sub><i>a</i></sub>(<i>x</i>) = <i>a</i> ⇒; <i>x</i>. Every <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> is a Heyting algebra.</li> <li><b><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></b>. A Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its <a href="/wiki/Transitive_reduction" title="Transitive reduction">transitive reduction</a>.</li> <li><b><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous relation</a></b>. A <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> on 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> </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> 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 X\times X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef859a7a6d98b98c35000664b8b0d5b2e094b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.447ex; height:2.176ex;" alt="{\displaystyle X\times X.}"></span> Said differently, it is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</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 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> and itself.</li></ul> <div class="mw-heading mw-heading2"><h2 id="I">I</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=9" title="Edit section: I"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></b>. An <b><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideal</a></b> is a subset <i>X</i> of a poset <i>P</i> that is a directed lower set. The dual notion is called <i>filter</i>.</li> <li><b><a href="/wiki/Incidence_algebra" title="Incidence algebra">Incidence algebra</a></b>. The <b><a href="/wiki/Incidence_algebra" title="Incidence algebra">incidence algebra</a></b> of a poset is the <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see <a href="/wiki/Incidence_algebra" title="Incidence algebra">incidence algebra</a> for the details.</li> <li><b><a href="/wiki/Infimum" class="mw-redirect" title="Infimum">Infimum</a></b>. For a poset <i>P</i> and a subset <i>X</i> of <i>P</i>, the greatest element in the set of lower bounds of <i>X</i> (if it exists, which it may not) is called the <b>infimum</b>, <b>meet</b>, or <b>greatest lower bound</b> of <i>X</i>. It is denoted by inf <i>X</i> 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 \bigwedge }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋀</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20eaee46377072a177e3577ea439d142574f6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigwedge }"></span><i>X</i>. The infimum of two elements may be written as inf{<i>x</i>,<i>y</i>} or <i>x</i> ∧ <i>y</i>. If the set <i>X</i> is finite, one speaks of a <b>finite infimum</b>. The dual notion is called <i>supremum</i>.</li> <li><b><a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">Interval</a></b>. For two elements <i>a</i>, <i>b</i> of a partially ordered set <i>P</i>, the <i>interval</i> [<i>a</i>,<i>b</i>] is the subset {<i>x</i> in <i>P</i> | <i>a</i> ≤ <i>x</i> ≤ <i>b</i>} of <i>P</i>. If <i>a</i> ≤ <i>b</i> does not hold the interval will be empty.</li> <li><span id="interval_finite_poset"></span><b>Interval finite poset</b>. A partially ordered set <i>P</i> is <b>interval finite</b> if every interval of the form {x in P | x ≤ a} is a finite set.<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></li> <li><b>Inverse</b>. See <i>converse</i>.</li> <li><b><a href="/wiki/Irreflexive" class="mw-redirect" title="Irreflexive">Irreflexive</a></b>. A <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a> <i>R</i> on a set <i>X</i> is irreflexive, if there is no element <i>x</i> in <i>X</i> such that <i>x R x</i>.</li> <li><b>Isotone</b>. See <i>monotone</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="J">J</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=10" title="Edit section: J"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Join</b>. See <i>supremum</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="L">L</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=11" title="Edit section: L"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></b>. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.</li> <li><b><a href="/wiki/Least_element" class="mw-redirect" title="Least element">Least element</a></b>. For a subset <i>X</i> of a poset <i>P</i>, an element <i>a</i> of <i>X</i> is called the least element of <i>X</i>, if <i>a</i> ≤ <i>x</i> for every element <i>x</i> in <i>X</i>. The dual notion is called <i>greatest element</i>.</li> <li>The <b>length</b> of a chain is the number of elements less one. A chain with 1 element has length 0, one with 2 elements has length 1, etc.</li> <li><b>Linear</b>. See <i>total order</i>.</li> <li><b><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></b>. A linear extension of a partial order is an extension that is a linear order, or total order.</li> <li><b><a href="/wiki/Complete_Heyting_algebra" title="Complete Heyting algebra">Locale</a></b>. A locale is a <i>complete Heyting algebra</i>. Locales are also called <i>frames</i> and appear in <a href="/wiki/Stone_duality" title="Stone duality">Stone duality</a> and <a href="/wiki/Pointless_topology" title="Pointless topology">pointless topology</a>.</li> <li><b><a href="/wiki/Locally_finite_poset" title="Locally finite poset">Locally finite poset</a></b>. A partially ordered set <i>P</i> is <i>locally finite</i> if every interval [<i>a</i>, <i>b</i>] = {<i>x</i> in <i>P</i> | <i>a</i> ≤ <i>x</i> ≤ <i>b</i>} is a finite set.</li> <li><b><a href="/wiki/Lower_bound" class="mw-redirect" title="Lower bound">Lower bound</a></b>. A lower bound of a subset <i>X</i> of a poset <i>P</i> is an element <i>b</i> of <i>P</i>, such that <i>b</i> ≤ <i>x</i>, for all <i>x</i> in <i>X</i>. The dual notion is called <i>upper bound</i>.</li> <li><b><a href="/wiki/Lower_set" class="mw-redirect" title="Lower set">Lower set</a></b>. A subset <i>X</i> of a poset <i>P</i> is called a lower set if, for all elements <i>x</i> in <i>X</i> and <i>p</i> in <i>P</i>, <i>p</i> ≤ <i>x</i> implies that <i>p</i> is contained in <i>X</i>. The dual notion is called <i>upper set</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="M">M</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=12" title="Edit section: M"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Maximal chain</b>. A <a href="/wiki/Total_order#Chains" title="Total order">chain</a> in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also excludes the existence of elements either less than all elements of the chain or greater than all its elements. A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset.</li> <li><b><a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">Maximal element</a></b>. A maximal element of a subset <i>X</i> of a poset <i>P</i> is an element <i>m</i> of <i>X</i>, such that <i>m</i> ≤ <i>x</i> implies <i>m</i> = <i>x</i>, for all <i>x</i> in <i>X</i>. The dual notion is called <i>minimal element</i>.</li> <li><b><a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">Maximum element</a></b>. Synonym of greatest element. For a subset <i>X</i> of a poset <i>P</i>, an element <i>a</i> of <i>X</i> is called the maximum element of <i>X</i> if <i>x</i> ≤ <i>a</i> for every element <i>x</i> in <i>X</i>. A maxim<i>um</i> element is necessarily maxim<i>al</i>, but the converse need not hold.</li> <li><b>Meet</b>. See <i>infimum</i>.</li> <li><b><a href="/wiki/Minimal_element" class="mw-redirect" title="Minimal element">Minimal element</a></b>. A minimal element of a subset <i>X</i> of a poset <i>P</i> is an element <i>m</i> of <i>X</i>, such that <i>x</i> ≤ <i>m</i> implies <i>m</i> = <i>x</i>, for all <i>x</i> in <i>X</i>. The dual notion is called <i>maximal element</i>.</li> <li><b><a href="/wiki/Least_element" class="mw-redirect" title="Least element">Minimum element</a></b>. Synonym of least element. For a subset <i>X</i> of a poset <i>P</i>, an element <i>a</i> of <i>X</i> is called the minimum element of <i>X</i> if <i>x</i> ≥ <i>a</i> for every element <i>x</i> in <i>X</i>. A minim<i>um</i> element is necessarily minim<i>al</i>, but the converse need not hold.</li> <li><b><a href="/wiki/Monotone_function" class="mw-redirect" title="Monotone function">Monotone</a></b>. A function <i>f</i> between posets <i>P</i> and <i>Q</i> is monotone if, for all elements <i>x</i>, <i>y</i> of <i>P</i>, <i>x</i> ≤ <i>y</i> (in <i>P</i>) implies <i>f</i>(<i>x</i>) ≤ <i>f</i>(<i>y</i>) (in <i>Q</i>). Other names for this property are <i>isotone</i> and <i>order-preserving</i>. In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, in the presence of <a href="/wiki/Total_order" title="Total order">total orders</a>, such functions are often called <b>monotonically increasing</b>, but this is not a very convenient description when dealing with non-total orders. The dual notion is called <i>antitone</i> or <i>order reversing</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="O">O</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=13" title="Edit section: O"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Order-dual</b>. The order dual of a partially ordered set is the same set with the partial order relation replaced by its converse.</li> <li><b><a href="/wiki/Order-embedding" class="mw-redirect" title="Order-embedding">Order-embedding</a></b>. A function <i>f</i> between posets <i>P</i> and <i>Q</i> is an order-embedding if, for all elements <i>x</i>, <i>y</i> of <i>P</i>, <i>x</i> ≤ <i>y</i> (in <i>P</i>) is equivalent to <i>f</i>(<i>x</i>) ≤ <i>f</i>(<i>y</i>) (in <i>Q</i>).</li> <li><b><a href="/wiki/Order_isomorphism" title="Order isomorphism">Order isomorphism</a></b>. A mapping <i>f</i>: <i>P</i> → <i>Q</i> between two posets <i>P</i> and <i>Q</i> is called an order isomorphism, if it is <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> and both <i>f</i> and <i>f</i><sup>−1</sup> are <a href="/wiki/Monotone_function" class="mw-redirect" title="Monotone function">monotone functions</a>. Equivalently, an order isomorphism is a surjective <i>order embedding</i>.</li> <li><b><a href="/wiki/Order-preserving" class="mw-redirect" title="Order-preserving">Order-preserving</a></b>. See <i>monotone</i>.</li> <li><b><a href="/wiki/Order-reversing" class="mw-redirect" title="Order-reversing">Order-reversing</a></b>. See <i>antitone</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="P">P</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=14" title="Edit section: P"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a></b>. A partial order is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> that is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>, <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>, and <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>. In a slight abuse of terminology, the term is sometimes also used to refer not to such a relation, but to its corresponding partially ordered set.</li> <li><b><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partially ordered set</a></b>. A partially ordered 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 (P,\leq ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4d603142d81c64ed0ad415c0fffed92039413e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.843ex;" alt="{\displaystyle (P,\leq ),}"></span> or <em>poset</em> for short, 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> together with 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"> <mspace width="thinmathspace" /> <mo>≤</mo> <mspace width="thinmathspace" /> </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/24112548985eab096493f73f838580442780b57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; 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 P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f4f085fcd14302f4f7a9bbdf77e816cccb3bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle P.}"></span></li> <li><b>Poset</b>. A partially ordered set.</li> <li><b><a href="/wiki/Preorder" title="Preorder">Preorder</a></b>. A preorder is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> that is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a> and <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>. Such orders may also be called <em>quasiorders</em> or <em>non-strict preorder</em>. The term <em>preorder</em> is also used to denote an <a href="#A">acyclic</a> <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> (also called an <em>acyclic digraph</em>).</li> <li><b><a href="/wiki/Preordered_set" class="mw-redirect" title="Preordered set">Preordered set</a></b>. A preordered 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 (P,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fe8855b84572c55012b0544255beb8d64b16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" alt="{\displaystyle (P,\leq )}"></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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> together with a preorder <span class="mwe-math-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> <mspace width="thinmathspace" /> </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/24112548985eab096493f73f838580442780b57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; 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 P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f4f085fcd14302f4f7a9bbdf77e816cccb3bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle P.}"></span></li> <li><b><a href="/wiki/Limit-preserving_function_(order_theory)" title="Limit-preserving function (order theory)">Preserving</a></b>. A function <i>f</i> between posets <i>P</i> and <i>Q</i> is said to preserve suprema (joins), if, for all subsets <i>X</i> of <i>P</i> that have a supremum sup <i>X</i> in <i>P</i>, we find that sup{<i>f</i>(<i>x</i>): <i>x</i> in <i>X</i>} exists and is equal to <i>f</i>(sup <i>X</i>). Such a function is also called <b>join-preserving</b>. Analogously, one says that <i>f</i> preserves finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called <i>join-reflecting</i>.</li> <li><b><a href="/wiki/Order_ideal" class="mw-redirect" title="Order ideal">Prime</a></b>. An <em>ideal</em> <i>I</i> in a lattice <i>L</i> is said to be prime, if, for all elements <i>x</i> and <i>y</i> in <i>L</i>, <i>x</i> ∧ <i>y</i> in <i>I</i> implies <i>x</i> in <i>I</i> or <i>y</i> in <i>I</i>. The dual notion is called a <i>prime filter</i>. Equivalently, a set is a prime filter <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> its complement is a prime ideal.</li> <li><b><a href="/wiki/Order_ideal" class="mw-redirect" title="Order ideal">Principal</a></b>. A filter is called <i>principal filter</i> if it has a least element. Dually, a <i>principal ideal</i> is an ideal with a greatest element. The least or greatest elements may also be called <i>principal elements</i> in these situations.</li> <li><b>Projection (operator)</b>. A self-map on a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> that is <a href="/wiki/Monotonic_function" title="Monotonic function">monotone</a> and <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a> under <a href="/wiki/Function_composition" title="Function composition">function composition</a>. Projections play an important role in <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>.</li> <li><b>Pseudo-complement</b>. In a <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a>, the element <i>x</i> ⇒; <i>0</i> is called the pseudo-complement of <i>x</i>. It is also given by sup{<i>y</i> : <i>y</i> ∧ <i>x</i> = 0}, i.e. as the least upper bound of all elements <i>y</i> with <i>y</i> ∧ <i>x</i> = 0.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Q">Q</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=15" title="Edit section: Q"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Quasiorder</b>. See <i>preorder</i>.</li> <li><b><a href="/wiki/Quasitransitive_relation" title="Quasitransitive relation">Quasitransitive</a></b>. A relation is quasitransitive if the relation on distinct elements is transitive. Transitive implies quasitransitive and quasitransitive implies acyclic.<sup id="cite_ref-BosSuz_1-1" class="reference"><a href="#cite_note-BosSuz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="R">R</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=16" title="Edit section: R"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Limit_preserving_(order_theory)" class="mw-redirect" title="Limit preserving (order theory)">Reflecting</a></b>. A function <i>f</i> between posets <i>P</i> and <i>Q</i> is said to reflect suprema (joins), if, for all subsets <i>X</i> of <i>P</i> for which the supremum sup{<i>f</i>(<i>x</i>): <i>x</i> in <i>X</i>} exists and is of the form <i>f</i>(<i>s</i>) for some <i>s</i> in <i>P</i>, then we find that sup <i>X</i> exists and that sup <i>X</i> = <i>s</i> . Analogously, one says that <i>f</i> reflects finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called <i>join-preserving</i>.</li> <li><b><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></b>. A <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> <i>R</i> on a set <i>X</i> is reflexive, if <i>x R x</i> holds for every element <i>x</i> in <i>X</i>.</li> <li><b>Residual</b>. A dual map attached to a <a href="/wiki/Residuated_mapping" title="Residuated mapping">residuated mapping</a>.</li> <li><b><a href="/wiki/Residuated_mapping" title="Residuated mapping">Residuated mapping</a></b>. A monotone map for which the preimage of a principal down-set is again principal. Equivalently, one component of a Galois connection.</li></ul> <div class="mw-heading mw-heading2"><h2 id="S">S</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=17" title="Edit section: S"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Saturated chain</b>. A <a href="/wiki/Total_order#Chains" title="Total order">chain</a> in a poset such that no element can be added <i>between two of its elements</i> without losing the property of being totally ordered. If the chain is finite, this means that in every pair of successive elements the larger one covers the smaller one. See also maximal chain.</li> <li><b><a href="/wiki/Scattered_order" title="Scattered order">Scattered</a></b>. A total order is scattered if it has no densely ordered subset.</li> <li><b><a href="/wiki/Scott-continuous" class="mw-redirect" title="Scott-continuous">Scott-continuous</a></b>. A monotone function <i>f</i> : <i>P</i> → <i>Q</i> between posets <i>P</i> and <i>Q</i> is Scott-continuous if, for every directed set <i>D</i> that has a supremum sup <i>D</i> in <i>P</i>, the set {<i>fx</i> | <i>x</i> in <i>D</i>} has the supremum <i>f</i>(sup <i>D</i>) in <i>Q</i>. Stated differently, a Scott-continuous function is one that <a href="/wiki/Limit_preserving_function_(order_theory)" class="mw-redirect" title="Limit preserving function (order theory)">preserves</a> all directed suprema. This is in fact equivalent to being <a href="/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">continuous</a> with respect to the <a href="/wiki/Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a> on the respective posets.</li> <li><b><a href="/wiki/Scott_domain" title="Scott domain">Scott domain</a></b>. A Scott domain is a partially ordered set which is a <a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">bounded complete</a> <a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">algebraic</a> <a href="/wiki/Complete_partial_order" title="Complete partial order">cpo</a>.</li> <li><b>Scott open</b>. See <i>Scott topology</i>.</li> <li><b>Scott topology</b>. For a poset <i>P</i>, a subset <i>O</i> is <b>Scott-open</b> if it is an <a href="/wiki/Upper_set" title="Upper set">upper set</a> and all directed sets <i>D</i> that have a supremum in <i>O</i> have non-empty intersection with <i>O</i>. The set of all Scott-open sets forms a <a href="/wiki/Topology" title="Topology">topology</a>, the <b>Scott topology</b>.</li> <li><b><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></b>. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-empty meets (infima) exist. Accordingly, one speaks of a <b>join-semilattice</b> or <b>meet-semilattice</b>.</li> <li><b>Smallest element</b>. See <i>least element</i>.</li> <li><a href="/wiki/Sperner_property_of_a_partially_ordered_set" title="Sperner property of a partially ordered set">Sperner property of a partially ordered set</a></li> <li><a href="/wiki/Sperner_poset" class="mw-redirect" title="Sperner poset">Sperner poset</a></li> <li><a href="/wiki/Strictly_Sperner_poset" class="mw-redirect" title="Strictly Sperner poset">Strictly Sperner poset</a></li> <li><a href="/wiki/Strongly_Sperner_poset" class="mw-redirect" title="Strongly Sperner poset">Strongly Sperner poset</a></li> <li><b><a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">Strict order</a></b>. See <em>strict partial order</em>.</li> <li><b><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict partial order</a></b>. A strict partial order is a <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous binary relation</a> that is <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, <a href="/wiki/Irreflexive_relation" class="mw-redirect" title="Irreflexive relation">irreflexive</a>, and <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>.</li> <li><b><a href="/wiki/Strict_preorder" class="mw-redirect" title="Strict preorder">Strict preorder</a></b>. See <em>strict partial order</em>.</li> <li><b><a href="/wiki/Supremum" class="mw-redirect" title="Supremum">Supremum</a></b>. For a poset <i>P</i> and a subset <i>X</i> of <i>P</i>, the <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a> in the set of <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bounds</a> of <i>X</i> (if it exists, which it may not) is called the <b>supremum</b>, <b>join</b>, or <b>least upper bound</b> of <i>X</i>. It is denoted by sup <i>X</i> 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 \bigvee }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋁</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigvee }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/424eb787b9be7b652deb858148ac5412c317aebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.582ex; height:3.843ex;" alt="{\displaystyle \bigvee }"></span><i>X</i>. The supremum of two elements may be written as sup{<i>x</i>,<i>y</i>} or <i>x</i> ∨ <i>y</i>. If the set <i>X</i> is finite, one speaks of a <b>finite supremum</b>. The dual notion is called <i>infimum</i>.</li> <li><b>Suzumura consistency</b>. A binary relation R is Suzumura consistent if <i>x</i> R<sup>∗</sup> <i>y</i> implies that <i>x</i> R <i>y</i> or not <i>y</i> R <i>x</i>.<sup id="cite_ref-BosSuz_1-2" class="reference"><a href="#cite_note-BosSuz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric relation</a></b>. A <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> <i>R</i> on a set <i>X</i> is symmetric, if <i>x R y</i> implies <i>y R x</i>, for all elements <i>x</i>, <i>y</i> in <i>X</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="T">T</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=18" title="Edit section: T"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Top</b>. See <i>unit</i>.</li> <li><b><a href="/wiki/Total_order" title="Total order">Total order</a></b>. A total order <i>T</i> is a partial order in which, for each <i>x</i> and <i>y</i> in <i>T</i>, we have <i>x</i> ≤ <i>y</i> or <i>y</i> ≤ <i>x</i>. Total orders are also called <i>linear orders</i> or <i>chains</i>.</li> <li><b><a href="/wiki/Connected_relation" title="Connected relation">Total relation</a></b>. Synonym for <i><a href="/wiki/Connected_relation" title="Connected relation">Connected relation</a></i>.</li> <li><b><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive relation</a></b>. A <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">relation</a> <i>R</i> on a set <i>X</i> is transitive, if <i>x R y</i> and <i>y R z</i> imply <i>x R z</i>, for all elements <i>x</i>, <i>y</i>, <i>z</i> in <i>X</i>.</li> <li><b><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></b>. The transitive closure R<sup>∗</sup> of a relation R consists of all pairs <i>x</i>,<i>y</i> for which there cists a finite chain <i>x</i> R <i>a</i>, <i>a</i> R <i>b</i>, ..., <i>z</i> R <i>y</i>.<sup id="cite_ref-BosSuz_1-3" class="reference"><a href="#cite_note-BosSuz-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="U">U</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=19" title="Edit section: U"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Unit</b>. The <a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">greatest element</a> of a poset <i>P</i> can be called <i>unit</i> or just <i>1</i> (if it exists). Another common term for this element is <b>top</b>. It is the infimum of the empty set and the supremum of <i>P</i>. The dual notion is called <i>zero</i>.</li> <li><b>Up-set</b>. See <i>upper set</i>.</li> <li><b><a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">Upper bound</a></b>. An upper bound of a subset <i>X</i> of a poset <i>P</i> is an element <i>b</i> of <i>P</i>, such that <i>x</i> ≤ <i>b</i>, for all <i>x</i> in <i>X</i>. The dual notion is called <i>lower bound</i>.</li> <li><b><a href="/wiki/Upper_set" title="Upper set">Upper set</a></b>. A subset <i>X</i> of a poset <i>P</i> is called an upper set if, for all elements <i>x</i> in <i>X</i> and <i>p</i> in <i>P</i>, <i>x</i> ≤ <i>p</i> implies that <i>p</i> is contained in <i>X</i>. The dual notion is called <i>lower set</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="V">V</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=20" title="Edit section: V"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Valuation</b>. Given 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 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>, a valuation <span class="mwe-math-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 \nu :X\to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu :X\to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef770a4099c2b4af5a9c34c51672a9ec042a3b4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.416ex; height:2.843ex;" alt="{\displaystyle \nu :X\to [0,1]}"></span> is strict (that is, <span class="mwe-math-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 \nu (\varnothing )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu (\varnothing )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d822864ae8d556078ff1769ee8b0f43402769505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.11ex; height:2.843ex;" alt="{\displaystyle \nu (\varnothing )=0}"></span>), monotone, modular (that is, <span class="mwe-math-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 \nu (U)+\nu (V)=\nu (U\cup V)+\nu (U\cap V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>∪</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ν</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 \nu (U)+\nu (V)=\nu (U\cup V)+\nu (U\cap V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/868d38b43d933e46dea5fa8ef7673b51eb293997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.82ex; height:2.843ex;" alt="{\displaystyle \nu (U)+\nu (V)=\nu (U\cup V)+\nu (U\cap V)}"></span>) and positive. Continuous valuations are a generalization of measures.</li></ul> <div class="mw-heading mw-heading2"><h2 id="W">W</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=21" title="Edit section: W"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Way-below_relation" class="mw-redirect" title="Way-below relation">Way-below relation</a></b>. In a poset <i>P</i>, some element <i>x</i> is <i>way below</i> <i>y</i>, written <i>x</i><<<i>y</i>, if for all directed subsets <i>D</i> of <i>P</i> which have a supremum, <i>y</i> ≤ <i>sup D</i> implies <i>x</i> ≤ <i>d</i> for some <i>d</i> in <i>D</i>. One also says that <i>x</i> <b>approximates</b> <i>y</i>. See also <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>.</li> <li><b><a href="/wiki/Weak_order" class="mw-redirect" title="Weak order">Weak order</a></b>. A partial order ≤ on a set <i>X</i> is a weak order provided that the poset (X, ≤) is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to a countable collection of sets ordered by comparison of <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Z">Z</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_order_theory&action=edit&section=22" title="Edit section: Z"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Zero</b>. The <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a> of a poset <i>P</i> can be called <i>zero</i> or just <i>0</i> (if it exists). Another common term for this element is <b>bottom</b>. Zero is the supremum of the empty set and the infimum of <i>P</i>. The dual notion is called <i>unit</i>.</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=Glossary_of_order_theory&action=edit&section=23" 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"><ol class="references"> <li id="cite_note-BosSuz-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-BosSuz_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BosSuz_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-BosSuz_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-BosSuz_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><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><cite id="CITEREFBossertSuzumura2010" class="citation book cs1">Bossert, Walter; Suzumura, Kōtarō (2010). <i>Consistency, choice and rationality</i>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0674052994" title="Special:BookSources/978-0674052994"><bdi>978-0674052994</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Consistency%2C+choice+and+rationality&rft.pub=Harvard+University+Press&rft.date=2010&rft.isbn=978-0674052994&rft.aulast=Bossert&rft.aufirst=Walter&rft.au=Suzumura%2C+K%C5%8Dtar%C5%8D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+order+theory" class="Z3988"></span></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="#CITEREFDeng2008">Deng 2008</a>, p. 22</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=Glossary_of_order_theory&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definitions given here are consistent with those that can be found in the following standard reference books: </p> <ul><li>B. A. Davey and H. A. Priestley, <i><a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order">Introduction to Lattices and Order</a></i>, 2nd Edition, Cambridge University Press, 2002.</li> <li>G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, <i>Continuous Lattices and Domains</i>, In <i>Encyclopedia of Mathematics and its Applications</i>, Vol. 93, Cambridge University Press, 2003.</li></ul> <p>Specific definitions: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeng2008" class="citation cs2">Deng, Bangming (2008), <i>Finite dimensional algebras and quantum groups</i>, Mathematical surveys and monographs, vol. 150, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4186-0" title="Special:BookSources/978-0-8218-4186-0"><bdi>978-0-8218-4186-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+dimensional+algebras+and+quantum+groups&rft.series=Mathematical+surveys+and+monographs&rft.pub=American+Mathematical+Society&rft.date=2008&rft.isbn=978-0-8218-4186-0&rft.aulast=Deng&rft.aufirst=Bangming&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGlossary+of+order+theory" class="Z3988"></span></li></ul> <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 .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><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 class="mw-selflink selflink">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 href="/wiki/Lattice_(order)" title="Lattice (order)">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 href="/wiki/Lattice_(order)" title="Lattice (order)">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><noscript><img 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