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Available in 21 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-21" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">21 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D8%B3%D9%85_%D9%87%D8%A7%D8%B3_%D8%A7%D9%84%D8%A8%D9%8A%D8%A7%D9%86%D9%8A" title="رسم هاس البياني – Arabic" lang="ar" hreflang="ar" data-title="رسم هاس البياني" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diagrama_de_Hasse" title="Diagrama de Hasse – Catalan" lang="ca" hreflang="ca" data-title="Diagrama de Hasse" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hasse%C5%AFv_diagram" title="Hasseův diagram – Czech" lang="cs" hreflang="cs" data-title="Hasseův diagram" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hasse-Diagramm" title="Hasse-Diagramm – German" lang="de" hreflang="de" data-title="Hasse-Diagramm" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diagrama_de_Hasse" title="Diagrama de Hasse – Spanish" lang="es" hreflang="es" data-title="Diagrama de Hasse" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%AF%D8%A7%D8%B1_%D9%87%D8%B3%D9%87" title="نمودار هسه – Persian" lang="fa" hreflang="fa" data-title="نمودار هسه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diagramme_de_Hasse" title="Diagramme de Hasse – French" lang="fr" hreflang="fr" data-title="Diagramme de Hasse" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%98%EC%84%B8_%EB%8F%84%ED%98%95" title="하세 도형 – Korean" lang="ko" hreflang="ko" data-title="하세 도형" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diagram_Hasse" title="Diagram Hasse – Indonesian" lang="id" hreflang="id" data-title="Diagram Hasse" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diagramma_di_Hasse" title="Diagramma di Hasse – Italian" lang="it" hreflang="it" data-title="Diagramma di Hasse" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%90%D7%92%D7%A8%D7%9E%D7%AA_%D7%94%D7%A1%D7%94" title="דיאגרמת הסה – Hebrew" lang="he" hreflang="he" data-title="דיאגרמת הסה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Diagramma_de_Hasse" title="Diagramma de Hasse – Lombard" lang="lmo" hreflang="lmo" data-title="Diagramma de Hasse" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hasse-diagram" title="Hasse-diagram – Hungarian" lang="hu" hreflang="hu" data-title="Hasse-diagram" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hasse-diagram" title="Hasse-diagram – Dutch" lang="nl" hreflang="nl" data-title="Hasse-diagram" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8F%E3%83%83%E3%82%BB%E5%9B%B3" title="ハッセ図 – Japanese" lang="ja" hreflang="ja" data-title="ハッセ図" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Diagram_Hassego" title="Diagram Hassego – Polish" lang="pl" hreflang="pl" data-title="Diagram Hassego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Diagrama_de_Hasse" title="Diagrama de Hasse – Portuguese" lang="pt" hreflang="pt" data-title="Diagrama de Hasse" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%BC%D0%B0_%D0%A5%D0%B0%D1%81%D1%81%D0%B5" title="Диаграмма Хассе – Russian" lang="ru" hreflang="ru" data-title="Диаграмма Хассе" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hassediagram" title="Hassediagram – Swedish" lang="sv" hreflang="sv" data-title="Hassediagram" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D0%B0%D0%B3%D1%80%D0%B0%D0%BC%D0%B0_%D0%93%D0%B0%D1%81%D1%81%D0%B5" title="Діаграма Гассе – Ukrainian" lang="uk" hreflang="uk" data-title="Діаграма Гассе" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%93%88%E6%96%AF%E5%9C%96" title="哈斯圖 – Chinese" lang="zh" hreflang="zh" data-title="哈斯圖" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Visual depiction of a partially ordered set</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Hess_diagram" title="Hess diagram">Hess diagram</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Inclusion_ordering.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Inclusion_ordering.svg/220px-Inclusion_ordering.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Inclusion_ordering.svg/330px-Inclusion_ordering.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Inclusion_ordering.svg/440px-Inclusion_ordering.svg.png 2x" data-file-width="250" data-file-height="200" /></a><figcaption>The <a href="/wiki/Power_set" title="Power set">power set</a> of a 2-element set ordered by <a href="/wiki/Subset" title="Subset">inclusion</a> </figcaption></figure> <p>In <a href="/wiki/Order_theory" title="Order theory">order theory</a>, a <b>Hasse diagram</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'h' in 'hi'">h</span><span title="/æ/: 'a' in 'bad'">æ</span><span title="'s' in 'sigh'">s</span><span title="/ə/: 'a' in 'about'">ə</span></span>/</a></span></span>; <style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">German:</span> <span class="IPA nowrap" lang="de-Latn-fonipa"><a href="/wiki/Help:IPA/Standard_German" title="Help:IPA/Standard German">[ˈhasə]</a></span>) is a type of <a href="/wiki/Mathematical_diagram" title="Mathematical diagram">mathematical diagram</a> used to represent a finite <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a>, in the form of a <a href="/wiki/Graph_drawing" title="Graph drawing">drawing</a> of its <a href="/wiki/Transitive_reduction" title="Transitive reduction">transitive reduction</a>. Concretely, for 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 (S,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e8fafe062081eef7510325dda0ea36c83a5a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.151ex; height:2.843ex;" alt="{\displaystyle (S,\leq )}"></span> one represents each element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> as a <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a> in the plane and draws a <a href="/wiki/Line_segment" title="Line segment">line segment</a> or curve that goes <i>upward</i> from one vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to another vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> <a href="/wiki/Covering_relation" title="Covering relation">covers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (that is, whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51b711ca7f932963cdb268b0817dc72d6258733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.676ex;" alt="{\displaystyle x\neq y}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and there is no <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> distinct from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq z\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>z</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq z\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc14a54fe48ebc7a2729afba14cc619270b0f0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.343ex;" alt="{\displaystyle x\leq z\leq y}"></span>). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. </p><p>Hasse diagrams are named after <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Helmut Hasse</a> (1898–1979); according to <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>, they are so called because of the effective use Hasse made of them.<sup id="cite_ref-FOOTNOTEBirkhoff1948_1-0" class="reference"><a href="#cite_note-FOOTNOTEBirkhoff1948-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in an 1895 work by Henri Gustave Vogt.<sup id="cite_ref-FOOTNOTEVogt1895_2-0" class="reference"><a href="#cite_note-FOOTNOTEVogt1895-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERival1985110_3-0" class="reference"><a href="#cite_note-FOOTNOTERival1985110-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using <a href="/wiki/Graph_drawing" title="Graph drawing">graph drawing</a> techniques.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In some sources, the phrase "Hasse diagram" has a different meaning: the <a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">directed acyclic graph</a> obtained from the covering relation of a partially ordered set, independently of any drawing of that graph.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Diagram_design">Diagram design</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hasse_diagram&action=edit&section=1" title="Edit section: Diagram design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Hasse diagrams are simple, as well as intuitive, tools for dealing with finite <a href="/wiki/Partially_ordered_set" title="Partially ordered set">posets</a>, it turns out to be rather difficult to draw "good" diagrams. The reason is that, in general, there are many different possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the <a href="/wiki/Minimal_element" class="mw-redirect" title="Minimal element">minimal elements</a> of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost. </p><p>The following example demonstrates the issue. Consider the <a href="/wiki/Power_set" title="Power set">power set</a> of a 4-element set ordered by inclusion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a924f8dcb2847bb8871edfdbf4c6b5cca0669228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \subseteq }"></span>. Below are four different Hasse diagrams for this partial order. Each subset has a node labelled with a binary encoding that shows whether a certain element is in the subset (1) or not (0): </p> <table style="margin: 0 auto;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Hypercubeorder_binary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Hypercubeorder_binary.svg/230px-Hypercubeorder_binary.svg.png" decoding="async" width="230" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Hypercubeorder_binary.svg/345px-Hypercubeorder_binary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Hypercubeorder_binary.svg/460px-Hypercubeorder_binary.svg.png 2x" data-file-width="360" data-file-height="360" /></a></span></td> <td>   </td> <td><span typeof="mw:File"><a href="/wiki/File:Hypercubecubes_binary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Hypercubecubes_binary.svg/260px-Hypercubecubes_binary.svg.png" decoding="async" width="260" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Hypercubecubes_binary.svg/390px-Hypercubecubes_binary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Hypercubecubes_binary.svg/520px-Hypercubecubes_binary.svg.png 2x" data-file-width="477" data-file-height="347" /></a></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Hypercubestar_binary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/240px-Hypercubestar_binary.svg.png" decoding="async" width="240" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/360px-Hypercubestar_binary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Hypercubestar_binary.svg/480px-Hypercubestar_binary.svg.png 2x" data-file-width="373" data-file-height="350" /></a></span> </td> <td>    </td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Hypercubematrix_binary.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Hypercubematrix_binary.svg/180px-Hypercubematrix_binary.svg.png" decoding="async" width="180" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Hypercubematrix_binary.svg/271px-Hypercubematrix_binary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Hypercubematrix_binary.svg/360px-Hypercubematrix_binary.svg.png 2x" data-file-width="251" data-file-height="319" /></a><figcaption></figcaption></figure> </td></tr></tbody></table> <p>The first diagram makes clear that the power set is a <a href="/wiki/Graded_poset" title="Graded poset">graded poset</a>. The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the <a href="/wiki/Tesseract" title="Tesseract">4-dimensional cube</a> is a combinatorial union of two 3-dimensional cubes, and that a tetrahedron (<a href="/wiki/Abstract_polytope" title="Abstract polytope">abstract 3-polytope</a>) likewise merges two triangles (<a href="/wiki/Abstract_polytope" title="Abstract polytope">abstract 2-polytopes</a>). The third diagram shows some of the internal symmetry of the structure. In the fourth diagram the vertices are arranged in a 4×4 grid. </p> <div class="mw-heading mw-heading2"><h2 id="Upward_planarity">Upward planarity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hasse_diagram&action=edit&section=2" title="Edit section: Upward planarity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Upward_planar_drawing" title="Upward planar drawing">Upward planar drawing</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dih4_subgroups.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Dih4_subgroups.svg/220px-Dih4_subgroups.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Dih4_subgroups.svg/330px-Dih4_subgroups.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Dih4_subgroups.svg/440px-Dih4_subgroups.svg.png 2x" data-file-width="729" data-file-height="675" /></a><figcaption>This Hasse diagram of the <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> of the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> <a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">Dih<sub>4</sub></a> has no crossing edges.</figcaption></figure> <p>If a partial order can be drawn as a Hasse diagram in which no two edges cross, its covering graph is said to be <i>upward planar</i>. A number of results on upward planarity and on crossing-free Hasse diagram construction are known: </p> <ul><li>If the partial order to be drawn is a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a>, then it can be drawn without crossings if and only if it has <a href="/wiki/Order_dimension" title="Order dimension">order dimension</a> at most two.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In this case, a non-crossing drawing may be found by deriving Cartesian coordinates for the elements from their positions in the two linear orders realizing the order dimension, and then rotating the drawing counterclockwise by a 45-degree angle.</li> <li>If the partial order has at most one <a href="/wiki/Minimal_element" class="mw-redirect" title="Minimal element">minimal element</a>, or it has at most one <a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">maximal element</a>, then it may be tested in <a href="/wiki/Linear_time" class="mw-redirect" title="Linear time">linear time</a> whether it has a non-crossing Hasse diagram.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>It is <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> to determine whether a partial order with multiple sources and sinks can be drawn as a crossing-free Hasse diagram.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, finding a crossing-free Hasse diagram is <a href="/wiki/Fixed-parameter_tractable" class="mw-redirect" title="Fixed-parameter tractable">fixed-parameter tractable</a> when parametrized by the number of <a href="/wiki/Articulation_point" class="mw-redirect" title="Articulation point">articulation points</a> and <a href="/wiki/Triconnected_component" class="mw-redirect" title="Triconnected component">triconnected components</a> of the transitive reduction of the partial order.<sup id="cite_ref-FOOTNOTEChan2004_9-0" class="reference"><a href="#cite_note-FOOTNOTEChan2004-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>If the <i>y</i>-coordinates of the elements of a partial order are specified, then a crossing-free Hasse diagram respecting those coordinate assignments can be found in linear time, if such a diagram exists.<sup id="cite_ref-FOOTNOTEJüngerLeipert1999_10-0" class="reference"><a href="#cite_note-FOOTNOTEJüngerLeipert1999-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In particular, if the input poset is a <a href="/wiki/Graded_poset" title="Graded poset">graded poset</a>, it is possible to determine in linear time whether there is a crossing-free Hasse diagram in which the height of each vertex is proportional to its rank.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Use_in_UML_notation">Use in UML notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hasse_diagram&action=edit&section=3" title="Edit section: Use in UML notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diamond_inheritance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Diamond_inheritance.svg/110px-Diamond_inheritance.svg.png" decoding="async" width="110" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Diamond_inheritance.svg/165px-Diamond_inheritance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Diamond_inheritance.svg/220px-Diamond_inheritance.svg.png 2x" data-file-width="180" data-file-height="270" /></a><figcaption>A <a href="/wiki/Class_diagram" title="Class diagram">class diagram</a> depicting <a href="/wiki/Multiple_inheritance" title="Multiple inheritance">multiple inheritance</a></figcaption></figure> <p>In <a href="/wiki/Software_engineering" title="Software engineering">software engineering</a> / <a href="/wiki/Object-oriented_design" class="mw-redirect" title="Object-oriented design">Object-oriented design</a>, the <a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">classes</a> of a software system and the <a href="/wiki/Inheritance_(object-oriented_programming)" title="Inheritance (object-oriented programming)">inheritance</a> relation between these classes is often depicted using a <a href="/wiki/Class_diagram" title="Class diagram">class diagram</a>, a form of Hasse diagram in which the edges connecting classes are drawn as solid line segments with an open triangle at the superclass end. </p> <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=Hasse_diagram&action=edit&section=4" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBirkhoff1948-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBirkhoff1948_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBirkhoff1948">Birkhoff (1948)</a>.</span> </li> <li id="cite_note-FOOTNOTEVogt1895-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVogt1895_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVogt1895">Vogt (1895)</a>.</span> </li> <li id="cite_note-FOOTNOTERival1985110-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERival1985110_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRival1985">Rival (1985)</a>, p. 110.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">E.g., see <a href="#CITEREFDi_BattistaTamassia1988">Di Battista & Tamassia (1988)</a> and <a href="#CITEREFFreese2004">Freese (2004)</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">For examples of this alternative meaning of Hasse diagrams, see <a href="#CITEREFChristofides1975">Christofides (1975</a>, pp. 170–174); <a href="#CITEREFThulasiramanSwamy1992">Thulasiraman & Swamy (1992)</a>; <a href="#CITEREFBang-Jensen2008">Bang-Jensen (2008)</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFGargTamassia1995a">Garg & Tamassia (1995a)</a>, Theorem 9, p. 118; <a href="#CITEREFBakerFishburnRoberts1971">Baker, Fishburn & Roberts (1971)</a>, theorem 4.1, page 18.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFGargTamassia1995a">Garg & Tamassia (1995a)</a>, Theorem 15, p. 125; <a href="#CITEREFBertolazziDi_BattistaManninoTamassia1993">Bertolazzi et al. (1993)</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFGargTamassia1995a">Garg & Tamassia (1995a)</a>, Corollary 1, p. 132; <a href="#CITEREFGargTamassia1995b">Garg & Tamassia (1995b)</a>.</span> </li> <li id="cite_note-FOOTNOTEChan2004-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEChan2004_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFChan2004">Chan (2004)</a>.</span> </li> <li id="cite_note-FOOTNOTEJüngerLeipert1999-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJüngerLeipert1999_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJüngerLeipert1999">Jünger & Leipert (1999)</a>.</span> </li> </ol></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=Hasse_diagram&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><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="CITEREFBakerFishburnRoberts1971" class="citation cs2">Baker, Kirby A.; <a href="/wiki/Peter_C._Fishburn" title="Peter C. Fishburn">Fishburn, Peter C.</a>; <a href="/wiki/Fred_S._Roberts" title="Fred S. Roberts">Roberts, Fred S.</a> (1971), "Partial orders of dimension 2", <i>Networks</i>, <b>2</b> (1): 11–28, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnet.3230020103">10.1002/net.3230020103</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Networks&rft.atitle=Partial+orders+of+dimension+2&rft.volume=2&rft.issue=1&rft.pages=11-28&rft.date=1971&rft_id=info%3Adoi%2F10.1002%2Fnet.3230020103&rft.aulast=Baker&rft.aufirst=Kirby+A.&rft.au=Fishburn%2C+Peter+C.&rft.au=Roberts%2C+Fred+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBang-Jensen2008" class="citation cs2">Bang-Jensen, Jørgen (2008), "2.1 Acyclic Digraphs", <i>Digraphs: Theory, Algorithms and Applications</i>, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, pp. 32–34, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84800-997-4" title="Special:BookSources/978-1-84800-997-4"><bdi>978-1-84800-997-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.1+Acyclic+Digraphs&rft.btitle=Digraphs%3A+Theory%2C+Algorithms+and+Applications&rft.series=Springer+Monographs+in+Mathematics&rft.pages=32-34&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=2008&rft.isbn=978-1-84800-997-4&rft.aulast=Bang-Jensen&rft.aufirst=J%C3%B8rgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBertolazziDi_BattistaManninoTamassia1993" class="citation cs2">Bertolazzi, R; 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<a href="/wiki/Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (1995a), "Upward planarity testing", <i><a href="/wiki/Order_(journal)" title="Order (journal)">Order</a></i>, <b>12</b> (2): 109–133, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01108622">10.1007/BF01108622</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14183717">14183717</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Order&rft.atitle=Upward+planarity+testing&rft.volume=12&rft.issue=2&rft.pages=109-133&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2FBF01108622&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14183717%23id-name%3DS2CID&rft.aulast=Garg&rft.aufirst=Ashim&rft.au=Tamassia%2C+Roberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGargTamassia1995b" class="citation cs2">Garg, Ashim; <a href="/wiki/Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (1995b), "On the computational complexity of upward and rectilinear planarity testing", <i><a href="/wiki/International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Graph Drawing (Proc. GD '94)</a></i>, LectureNotes in Computer Science, vol. 894, Springer-Verlag, pp. 286–297, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-58950-3_384">10.1007/3-540-58950-3_384</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58950-1" title="Special:BookSources/978-3-540-58950-1"><bdi>978-3-540-58950-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+computational+complexity+of+upward+and+rectilinear+planarity+testing&rft.btitle=Graph+Drawing+%28Proc.+GD+%2794%29&rft.series=LectureNotes+in+Computer+Science&rft.pages=286-297&rft.pub=Springer-Verlag&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2F3-540-58950-3_384&rft.isbn=978-3-540-58950-1&rft.aulast=Garg&rft.aufirst=Ashim&rft.au=Tamassia%2C+Roberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJüngerLeipert1999" class="citation cs2">Jünger, Michael; Leipert, Sebastian (1999), "Level planar embedding in linear time", <i><a href="/wiki/International_Symposium_on_Graph_Drawing" title="International Symposium on Graph Drawing">Graph Drawing (Proc. GD '99)</a></i>, Lecture Notes in Computer Science, vol. 1731, pp. 72–81, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-46648-7_7">10.1007/3-540-46648-7_7</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66904-3" title="Special:BookSources/978-3-540-66904-3"><bdi>978-3-540-66904-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Level+planar+embedding+in+linear+time&rft.btitle=Graph+Drawing+%28Proc.+GD+%2799%29&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=72-81&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2F3-540-46648-7_7&rft.isbn=978-3-540-66904-3&rft.aulast=J%C3%BCnger&rft.aufirst=Michael&rft.au=Leipert%2C+Sebastian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRival1985" class="citation cs2">Rival, Ivan (1985), "The diagram", in Rival, Ivan (ed.), <i>Graphs and Order: The Role of Graphs in the Theory of Ordered Sets and Its Applications, Proceedings of the NATO Advanced Study Institute held in Banff, May 18–31, 1984</i>, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 147, Reidel, Dordrecht, pp. 103–133, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0818494">0818494</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+diagram&rft.btitle=Graphs+and+Order%3A+The+Role+of+Graphs+in+the+Theory+of+Ordered+Sets+and+Its+Applications%2C+Proceedings+of+the+NATO+Advanced+Study+Institute+held+in+Banff%2C+May+18%E2%80%9331%2C+1984&rft.series=NATO+Advanced+Science+Institutes+Series+C%3A+Mathematical+and+Physical+Sciences&rft.pages=103-133&rft.pub=Reidel%2C+Dordrecht&rft.date=1985&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D818494%23id-name%3DMR&rft.aulast=Rival&rft.aufirst=Ivan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThulasiramanSwamy1992" class="citation cs2">Thulasiraman, K.; Swamy, M. N. S. (1992), "5.7 Acyclic Directed Graphs", <i>Graphs: Theory and Algorithms</i>, John Wiley and Son, p. 118, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-51356-8" title="Special:BookSources/978-0-471-51356-8"><bdi>978-0-471-51356-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=5.7+Acyclic+Directed+Graphs&rft.btitle=Graphs%3A+Theory+and+Algorithms&rft.pages=118&rft.pub=John+Wiley+and+Son&rft.date=1992&rft.isbn=978-0-471-51356-8&rft.aulast=Thulasiraman&rft.aufirst=K.&rft.au=Swamy%2C+M.+N.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVogt1895" class="citation cs2">Vogt, Henri Gustave (1895), <i>Leçons sur la résolution algébrique des équations</i>, Nony, p. 91</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le%C3%A7ons+sur+la+r%C3%A9solution+alg%C3%A9brique+des+%C3%A9quations&rft.pages=91&rft.pub=Nony&rft.date=1895&rft.aulast=Vogt&rft.aufirst=Henri+Gustave&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hasse_diagram&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Related media at Wikimedia Commons: <ul><li><a href="https://commons.wikimedia.org/wiki/Hasse_diagram" class="extiw" title="commons:Hasse diagram">Hasse diagram</a> (Gallery)</li> <li><a href="https://commons.wikimedia.org/wiki/Category:Hasse_diagrams" class="extiw" title="commons:Category:Hasse diagrams">Hasse diagrams</a> (Category)</li></ul></li> <li><span class="citation mathworld" id="Reference-Mathworld-Hasse_Diagram"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HasseDiagram.html">"Hasse Diagram"</a>, <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Hasse+Diagram&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHasseDiagram.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHasse+diagram" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><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 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href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a 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 class="mw-selflink selflink">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 src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hasse_diagram&oldid=1241269355">https://en.wikipedia.org/w/index.php?title=Hasse_diagram&oldid=1241269355</a>"</div></div> <div id="catlinks" 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