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href="https://es.wikipedia.org/wiki/Secci%C3%B3n_final" title="Sección final – Spanish" lang="es" hreflang="es" data-title="Sección final" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Section_commen%C3%A7ante" title="Section commençante – French" lang="fr" hreflang="fr" data-title="Section commençante" 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/%EC%83%81%EC%A7%91%ED%95%A9" title="상집합 – Korean" lang="ko" hreflang="ko" data-title="상집합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a 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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">Subset of a preorder that contains all larger elements</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Upset_210div.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Upset_210div.svg/220px-Upset_210div.svg.png" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Upset_210div.svg/330px-Upset_210div.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Upset_210div.svg/440px-Upset_210div.svg.png 2x" data-file-width="528" data-file-height="415" /></a><figcaption>A <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of the <a href="/wiki/Divisor" title="Divisor">divisors</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 210}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>210</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 210}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6671b800d11c9d83a8d3981fbd1765237e5530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 210}"></span>, ordered by the relation <i>is divisor of</i>, with the upper 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 \uparrow 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9f40389a3ec6a3293311f8919a76d71b9818ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.97ex; height:2.509ex;" alt="{\displaystyle \uparrow 2}"></span> colored green. The white sets form the lower 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 \downarrow 105.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mn>105.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow 105.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6488db838b1145c9c1d76c1e28dd0854f24aabcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.942ex; height:2.509ex;" alt="{\displaystyle \downarrow 105.}"></span></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>upper set</b> (also called an <b>upward closed set</b>, an <b>upset</b>, or an <b>isotone</b> set in <i>X</i>)<sup id="cite_ref-FOOTNOTEDoleckiMynard201627–29_1-0" class="reference"><a href="#cite_note-FOOTNOTEDoleckiMynard201627–29-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> of a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</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,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"></span> with the following property: if <i>s</i> is in <i>S</i> and if <i>x</i> in <i>X</i> is larger than <i>s</i> (that is, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s<x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9961558e14acd5c2316c30009931645bb1da8442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.519ex; height:1.843ex;" alt="{\displaystyle s<x}"></span>), then <i>x</i> is in <i>S</i>. In other words, this means that any <i>x</i> element of <i>X</i> 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 \,\geq \,}"> <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 \,\geq \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9eb9fa17eaf66723084db056ce4c88ba93110cc" 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 \,\geq \,}"></span> to some element of <i>S</i> is necessarily also an element of <i>S</i>. The term <b>lower set</b> (also called a <b>downward closed set</b>, <b>down set</b>, <b>decreasing set</b>, <b>initial segment</b>, or <b>semi-ideal</b>) is defined similarly as being a subset <i>S</i> of <i>X</i> with the property that any element <i>x</i> of <i>X</i> 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 \,\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> to some element of <i>S</i> is necessarily also an element of <i>S</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Upper_set&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> be a <a href="/wiki/Preordered_set" class="mw-redirect" title="Preordered set">preordered set</a>. An <b><em>upper set</em></b> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> (also called an <b><em>upward closed set</em></b>, an <b><em>upset</em></b>, or an <b><em>isotone</em> set</b>)<sup id="cite_ref-FOOTNOTEDoleckiMynard201627–29_1-1" class="reference"><a href="#cite_note-FOOTNOTEDoleckiMynard201627–29-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}"></span> that is "closed under going up", in the sense that </p> <dl><dd>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c3e0203fca1c3b6c1982520132b9f057ed5d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle u\in U}"></span> and all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in 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\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80edd347954a2e6eef6e88800f4ac8d7bc75d8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.758ex; height:2.176ex;" alt="{\displaystyle u\leq x}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2494bed786064aac20efd5ea0fcaa3f787dd6f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.6ex; height:2.176ex;" alt="{\displaystyle x\in U.}"></span></dd></dl> <p>The <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual</a> notion is a <b><em>lower set</em></b> (also called a <b><em>downward closed set</em></b>, <b><em>down set</em></b>, <b><em>decreasing set</em></b>, <b><em>initial segment</em></b>, or <b><em>semi-ideal</em></b>), which is a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd53b3a46c486f5d516a6b0d8298b819c964044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.661ex; height:2.343ex;" alt="{\displaystyle L\subseteq X}"></span> that is "closed under going down", in the sense that </p> <dl><dd>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\in L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>∈</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\in L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af31b518b8600dc8ff15041dc353769574d4631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.117ex; height:2.176ex;" alt="{\displaystyle l\in L}"></span> and all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in 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\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a570d97ebf62f73525844ee24a254a0309ba24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.121ex; height:2.343ex;" alt="{\displaystyle x\leq l}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in L.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>L</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in L.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d701f876b80892acff3cc9387f1524702bc20ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.4ex; height:2.176ex;" alt="{\displaystyle x\in L.}"></span></dd></dl> <p>The terms <b><em>order ideal</em></b> or <b><em><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideal</a></em></b> are sometimes used as synonyms for lower set.<sup id="cite_ref-DP_2-0" class="reference"><a href="#cite_note-DP-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> This choice of terminology fails to reflect the notion of an ideal of a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> because a lower set of a lattice is not necessarily a sublattice.<sup id="cite_ref-DP_2-1" class="reference"><a href="#cite_note-DP-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Upper_set&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Every partially ordered set is an upper set of itself.</li> <li>The <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> and the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of any family of upper sets is again an upper set.</li> <li>The <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> of any upper set is a lower set, and vice versa.</li> <li>Given 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 (X,\leq ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8245e7b4904423fcdc9e821d5f6510e70dd4c4ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.278ex; height:2.843ex;" alt="{\displaystyle (X,\leq ),}"></span> the family of upper sets of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </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> ordered with the <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusion</a> relation is a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a>, the <b>upper set lattice</b>.</li> <li>Given an arbitrary subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> of 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 X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}"></span> the smallest upper set containing <span class="mwe-math-element"><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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is denoted using an up arrow as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffdfe5056b42cc6f405fd7ae90fd94e2a782b554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.581ex; height:2.509ex;" alt="{\displaystyle \uparrow Y}"></span> (see <a href="#Upper_closure_and_lower_closure">upper closure and lower closure</a>). <ul><li>Dually, the smallest lower set containing <span class="mwe-math-element"><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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is denoted using a down arrow as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3694f6f55642adb6af95a8b528eb8911c47940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.228ex; height:2.509ex;" alt="{\displaystyle \downarrow Y.}"></span></li></ul></li> <li>A lower set is called <b>principal</b> if it is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow \{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fc2a373930a2f06f1e4ecd7d6eeea215881ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.462ex; height:2.843ex;" alt="{\displaystyle \downarrow \{x\}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is an 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 X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}"></span></li> <li>Every lower 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 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> of a finite 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 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 equal to the smallest lower set containing all <a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">maximal elements</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mi>Y</mi> <mo>=↓</mo> <mi>Max</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b884dbf0cf7dee5ca8476f1af5905977b18c13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.945ex; height:2.843ex;" alt="{\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Max} (Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Max</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Max} (Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6f7fed2e76ff2cbfb0bf49cb515363b6f54082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.104ex; height:2.843ex;" alt="{\displaystyle \operatorname {Max} (Y)}"></span> denotes the set containing the maximal elements 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 Y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle Y.}"></span></li></ul></li> <li>A <a href="/wiki/Directed_set" title="Directed set">directed</a> lower set is called an <a href="/wiki/Order_ideal" class="mw-redirect" title="Order ideal">order ideal</a>.</li> <li>For partial orders satisfying the <a href="/wiki/Descending_chain_condition" class="mw-redirect" title="Descending chain condition">descending chain condition</a>, antichains and upper sets are in one-to-one correspondence via the following <a href="/wiki/Bijections" class="mw-redirect" title="Bijections">bijections</a>: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets of <a href="/wiki/Real_number" title="Real number">real numbers</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\in \mathbb {R} :x>0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} :x>0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6064b0a0b726ab48094164b2f82a9d34cf640fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.701ex; height:2.843ex;" alt="{\displaystyle \{x\in \mathbb {R} :x>0\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} :x>1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>></mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in \mathbb {R} :x>1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98446b0ed9a381b51059aa67f192f0cd62453f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.701ex; height:2.843ex;" alt="{\displaystyle \{x\in \mathbb {R} :x>1\}}"></span> are both mapped to the empty antichain.</li></ul> <p><span class="anchor" id="Upper_closure"></span><span class="anchor" id="Upward_closure"></span><span class="anchor" id="Lower_closure"></span><span class="anchor" id="Downward_closure"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Upper_closure_and_lower_closure">Upper closure and lower closure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Upper_set&action=edit&section=3" title="Edit section: Upper closure and lower closure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of 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 (X,\leq ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8245e7b4904423fcdc9e821d5f6510e70dd4c4ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.278ex; height:2.843ex;" alt="{\displaystyle (X,\leq ),}"></span> the <b>upper closure</b> or <b>upward closure</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\uparrow X},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> <mi>X</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\uparrow X},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d73f69efd592e433d627b059a61a287b0e34b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.431ex; height:3.009ex;" alt="{\displaystyle x^{\uparrow X},}"></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^{\uparrow },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\uparrow },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5555fc04c944941c4d0378a707990c865d469c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:3.009ex;" alt="{\displaystyle x^{\uparrow },}"></span> 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 \uparrow \!x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow \!x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351a3fb4a904d8ca6d9675996f1a1c329efea76f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.397ex; height:2.509ex;" alt="{\displaystyle \uparrow \!x,}"></span> is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> <mi>X</mi> </mrow> </msup> <mo>=</mo> <mspace width="thickmathspace" /> <mo stretchy="false">↑</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>x</mi> <mo>≤</mo> <mi>u</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d50b5f926d431a68e520a84589bdb5b926ba318" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.901ex; height:3.176ex;" alt="{\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}}"></span> while the <b>lower closure</b> or <b>downward closure</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </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>, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\downarrow X},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> <mi>X</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\downarrow X},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb42804e594b656735eeaaa9badf9b90712eadf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.431ex; height:3.009ex;" alt="{\displaystyle x^{\downarrow X},}"></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^{\downarrow },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\downarrow },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86ae7dbed010a0bfc366475c05d206ed5bf5182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:3.009ex;" alt="{\displaystyle x^{\downarrow },}"></span> 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 \downarrow \!x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow \!x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ad91f02f32b174ba545b6dfbfb2c03276f6b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.397ex; height:2.509ex;" alt="{\displaystyle \downarrow \!x,}"></span> is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> <mi>X</mi> </mrow> </msup> <mo>=</mo> <mspace width="thickmathspace" /> <mo stretchy="false">↓</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>l</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>l</mi> <mo>≤</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd397d9a7135d9ce2f5940b403816d7ba342f668" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.275ex; height:3.176ex;" alt="{\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}"></span> </p><p>The 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 \uparrow \!x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow \!x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9586b0b6264416c5574e3e7da11afb9486467e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle \uparrow \!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 \downarrow \!x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow \!x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234e75b86db7d3e610c9b64f8f90d63991786d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle \downarrow \!x}"></span> are, respectively, the smallest upper and lower sets containing <span class="mwe-math-element"><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> as an element. More generally, given a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a030c36801bc01950928058517e7a7da7eaa08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.468ex; height:2.509ex;" alt="{\displaystyle A\subseteq X,}"></span> define the <b>upper</b>/<b>upward closure</b> and the <b>lower</b>/<b>downward closure</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"></span> denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\uparrow X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\uparrow X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cde1a024c6d238bca34d112241173882a9ef1e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.197ex; height:2.676ex;" alt="{\displaystyle A^{\uparrow 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 A^{\downarrow X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> <mi>X</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\downarrow X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0787118c6acd2c083c2b768e95c6ba89432c47ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.197ex; height:2.676ex;" alt="{\displaystyle A^{\downarrow X}}"></span> respectively, as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> <mi>X</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mo stretchy="false">↑</mo> <mspace width="negativethinmathspace" /> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb81b6c4e020020ef711ddd9a08461d6087c8d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.686ex; height:5.676ex;" alt="{\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> <mi>X</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <mo stretchy="false">↓</mo> <mspace width="negativethinmathspace" /> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed1ab05ad7249651b4d055629360fa518295a53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.332ex; height:5.676ex;" alt="{\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}"></span> </p><p>In this way, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow x=\uparrow \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> <mi>x</mi> <mo>=↑</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow x=\uparrow \{x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/441300e6aa05d4af8916e959b4c73f1f4bfc3444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.053ex; height:2.843ex;" alt="{\displaystyle \uparrow x=\uparrow \{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 \downarrow x=\downarrow \{x\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> <mi>x</mi> <mo>=↓</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow x=\downarrow \{x\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23f60ae2283b95417ebba50bce04e748ceb12f9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.7ex; height:2.843ex;" alt="{\displaystyle \downarrow x=\downarrow \{x\},}"></span> where upper sets and lower sets of this form are called <b>principal</b>. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it. </p><p>The upper and lower closures, when viewed as functions from the power set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </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> to itself, are examples of <a href="/wiki/Kuratowski_closure_axioms#Definition" title="Kuratowski closure axioms">closure operators</a> since they satisfy all of the <a href="/wiki/Kuratowski_closure_axioms" title="Kuratowski closure axioms">Kuratowski closure axioms</a>. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. (Indeed, this is a general phenomenon of closure operators. For example, the <a href="/wiki/Topological_closure" class="mw-redirect" title="Topological closure">topological closure</a> of a set is the intersection of all <a href="/wiki/Closed_sets" class="mw-redirect" title="Closed sets">closed sets</a> containing it; the <a href="/wiki/Linear_span" title="Linear span">span</a> of a set of vectors is the intersection of all <a href="/wiki/Linear_subspace" title="Linear subspace">subspaces</a> containing it; the <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">subgroup generated by a subset</a> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is the intersection of all subgroups containing it; the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by a subset of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> is the intersection of all ideals containing it; and so on.) </p> <div class="mw-heading mw-heading2"><h2 id="Ordinal_numbers">Ordinal numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Upper_set&action=edit&section=4" title="Edit section: Ordinal numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Upper_set&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Abstract_simplicial_complex" title="Abstract simplicial complex">Abstract simplicial complex</a> (also called: <a href="/wiki/Independence_system" title="Independence system">Independence system</a>) - a set-family that is downwards-closed with respect to the containment relation.</li> <li><a href="/wiki/Cofinal_set" class="mw-redirect" title="Cofinal set">Cofinal set</a> – a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> of 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 (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> that contains for every element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in 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\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ebdb0a09f0721ccdd0b779e0a21caf386be82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}"></span> some element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq 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\leq y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6616694036cf80c75a6bf94273b2e902cfc41884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.343ex;" alt="{\displaystyle x\leq y.}"></span></li></ul> <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=Upper_set&action=edit&section=6" title="Edit section: References"><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-FOOTNOTEDoleckiMynard201627–29-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDoleckiMynard201627–29_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDoleckiMynard201627–29_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDoleckiMynard2016">Dolecki & Mynard 2016</a>, pp. 27–29.</span> </li> <li id="cite_note-DP-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-DP_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DP_2-1"><sup><i><b>b</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="CITEREFBrian_A._DaveyHilary_Ann_Priestley2002" class="citation book cs1">Brian A. Davey; <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">Hilary Ann Priestley</a> (2002). <a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order"><i>Introduction to Lattices and Order</i></a> (2nd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 20, 44. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-78451-4" title="Special:BookSources/0-521-78451-4"><bdi>0-521-78451-4</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2001043910">2001043910</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Lattices+and+Order&rft.pages=20%2C+44&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2002&rft_id=info%3Alccn%2F2001043910&rft.isbn=0-521-78451-4&rft.au=Brian+A.+Davey&rft.au=Hilary+Ann+Priestley&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUpper+set" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley2002" class="citation book cs1">Stanley, R.P. (2002). <i>Enumerative combinatorics</i>. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-66351-9" title="Special:BookSources/978-0-521-66351-9"><bdi>978-0-521-66351-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+combinatorics&rft.series=Cambridge+studies+in+advanced+mathematics&rft.pages=100&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-66351-9&rft.aulast=Stanley&rft.aufirst=R.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUpper+set" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawson1998" class="citation book cs1">Lawson, M.V. (1998). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/inversesemigroup00laws"><i>Inverse semigroups: the theory of partial symmetries</i></a></span>. World Scientific. p. <a rel="nofollow" class="external text" href="https://archive.org/details/inversesemigroup00laws/page/n34">22</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-3316-7" title="Special:BookSources/978-981-02-3316-7"><bdi>978-981-02-3316-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inverse+semigroups%3A+the+theory+of+partial+symmetries&rft.pages=22&rft.pub=World+Scientific&rft.date=1998&rft.isbn=978-981-02-3316-7&rft.aulast=Lawson&rft.aufirst=M.V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Finversesemigroup00laws&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUpper+set" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanck,_J.2000" class="citation journal cs1">Blanck, J. (2000). <a rel="nofollow" class="external text" href="http://www-compsci.swan.ac.uk/~csjens/pdf/top.pdf">"Domain representations of topological spaces"</a> <span class="cs1-format">(PDF)</span>. <i>Theoretical Computer Science</i>. <b>247</b> (<span class="nowrap">1–</span>2): <span class="nowrap">229–</span>255. <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/10.1016%2Fs0304-3975%2899%2900045-6">10.1016/s0304-3975(99)00045-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Domain+representations+of+topological+spaces&rft.volume=247&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E255&rft.date=2000&rft_id=info%3Adoi%2F10.1016%2Fs0304-3975%2899%2900045-6&rft.au=Blanck%2C+J.&rft_id=http%3A%2F%2Fwww-compsci.swan.ac.uk%2F~csjens%2Fpdf%2Ftop.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUpper+set" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoleckiMynard2016" class="citation book cs1"><a href="/w/index.php?title=Szymon_Dolecki&action=edit&redlink=1" class="new" title="Szymon Dolecki (page does not exist)">Dolecki, Szymon</a>; Mynard, Frédéric (2016). <i>Convergence Foundations Of Topology</i>. New Jersey: World Scientific Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4571-52-4" title="Special:BookSources/978-981-4571-52-4"><bdi>978-981-4571-52-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/945169917">945169917</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convergence+Foundations+Of+Topology&rft.place=New+Jersey&rft.pub=World+Scientific+Publishing+Company&rft.date=2016&rft_id=info%3Aoclcnum%2F945169917&rft.isbn=978-981-4571-52-4&rft.aulast=Dolecki&rft.aufirst=Szymon&rft.au=Mynard%2C+Fr%C3%A9d%C3%A9ric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUpper+set" class="Z3988"></span></li> <li>Hoffman, K. H. (2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070621125416/http://www.mathematik.tu-darmstadt.de:8080/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf"><i>The low separation axioms (T<sub>0</sub>) and (T<sub>1</sub>)</i></a></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 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 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class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a 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_theory153" 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 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 class="mw-selflink selflink">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?useformat=desktop&type=1x1&usesul3=0" 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=Upper_set&oldid=1136854557">https://en.wikipedia.org/w/index.php?title=Upper_set&oldid=1136854557</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Order theory</a></li></ul></div><div id="mw-hidden-catlinks" 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