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class="vector-toc-numb">2</span> <span>Axiomatics</span> </div> </a> <ul id="toc-Axiomatics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_other_kinds_of_order" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_other_kinds_of_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relation to other kinds of order</span> </div> </a> <button aria-controls="toc-Relation_to_other_kinds_of_order-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation to other kinds of order subsection</span> </button> <ul id="toc-Relation_to_other_kinds_of_order-sublist" class="vector-toc-list"> <li id="toc-Partial_orders" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Partial orders</span> </div> </a> <ul id="toc-Partial_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_orders" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Weak orders</span> </div> </a> <ul id="toc-Weak_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interval_orders" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interval_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Interval orders</span> </div> </a> <ul id="toc-Interval_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quasitransitive_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quasitransitive_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Quasitransitive relations</span> </div> </a> <ul id="toc-Quasitransitive_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Combinatorial_enumeration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Combinatorial_enumeration"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Combinatorial enumeration</span> </div> </a> <ul id="toc-Combinatorial_enumeration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other results</span> </div> </a> <ul id="toc-Other_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Semiorder</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="This article exist only in this language. 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Semiorder.svg/255px-Semiorder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Semiorder.svg/340px-Semiorder.svg.png 2x" data-file-width="216" data-file-height="252" /></a><figcaption>The <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of a semiorder. Two items are comparable when their vertical coordinates differ by at least one unit (the spacing between solid blue lines).</figcaption></figure> <p>In <a href="/wiki/Order_theory" title="Order theory">order theory</a>, a branch of mathematics, a <b>semiorder</b> is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given <a href="/wiki/Margin_of_error" title="Margin of error">margin of error</a> are deemed <a href="/wiki/Comparability" title="Comparability">incomparable</a>. Semiorders were introduced and applied in <a href="/wiki/Mathematical_psychology" title="Mathematical psychology">mathematical psychology</a> by <a href="/wiki/R._Duncan_Luce" title="R. Duncan Luce">Duncan Luce</a> (<a href="#CITEREFLuce1956">1956</a>) as a model of human preference. They generalize <a href="/wiki/Strict_weak_ordering" class="mw-redirect" title="Strict weak ordering">strict weak orderings</a>, in which items with equal scores may be tied but there is no margin of error. They are a special case of <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial orders</a> and of <a href="/wiki/Interval_order" title="Interval order">interval orders</a>, and can be characterized among the partial orders by additional <a href="/wiki/Axiom" title="Axiom">axioms</a>, or by two forbidden four-item suborders. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Utility_theory">Utility theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=1" title="Edit section: Utility theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a <a href="/wiki/Transitive_relation" title="Transitive relation">transitive relation</a>. For instance, suppose 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}"> <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>, <span class="mwe-math-element"><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>, 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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> represent three quantities of the same material, and 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}"> <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 larger than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> by the smallest amount that is perceptible as a difference, while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is halfway between the two of them. Then, a person who desires more of the material would prefer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>, but would not have a preference between the other two pairs. In this example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are incomparable in the preference ordering, as are <span class="mwe-math-element"><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> 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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> are comparable, so incomparability does not obey the transitive law.<sup id="cite_ref-FOOTNOTELuce1956179_1-0" class="reference"><a href="#cite_note-FOOTNOTELuce1956179-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>To model this mathematically, suppose that objects are given numerical <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility values</a>, by letting <span class="mwe-math-element"><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/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> be any <a href="/wiki/Utility_function" class="mw-redirect" title="Utility function">utility function</a> that maps the objects to be compared (a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>) to <a href="/wiki/Real_number" title="Real number">real numbers</a>. Set a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each other are declared incomparable, and define a binary relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> on the objects, by setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)\leq u(y)-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x)\leq u(y)-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68521cc539c0427404882b64f57c93e92f67d3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.864ex; height:2.843ex;" alt="{\displaystyle u(x)\leq u(y)-1}"></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,<)}"> <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,<)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06a913707358013e41f176121fbfc0c1e3c78ef" 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,<)}"></span> forms a semiorder.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> If, instead, objects are declared comparable whenever their utilities differ, the result would be a <a href="/wiki/Strict_weak_order" class="mw-redirect" title="Strict weak order">strict weak ordering</a>, for which incomparability of objects (based on equality of numbers) would be transitive.<sup id="cite_ref-FOOTNOTELuce1956179_1-1" class="reference"><a href="#cite_note-FOOTNOTELuce1956179-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Axiomatics">Axiomatics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=2" title="Edit section: Axiomatics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:200px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Suborder_2.svg" class="mw-file-description"><img alt="Two mutually incomparable two-point linear orders" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Suborder_2.svg/142px-Suborder_2.svg.png" decoding="async" width="142" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Suborder_2.svg/213px-Suborder_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Suborder_2.svg/284px-Suborder_2.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></div><div class="thumbcaption">Forbidden: two mutually incomparable two-point <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linear orders</a></div></div><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:200px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Suborder_1.svg" class="mw-file-description"><img alt="A three-point linear order, with a fourth incomparable point" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Suborder_1.svg/142px-Suborder_1.svg.png" decoding="async" width="142" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Suborder_1.svg/213px-Suborder_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Suborder_1.svg/284px-Suborder_1.svg.png 2x" data-file-width="744" data-file-height="1052" /></a></span></div><div class="thumbcaption">Forbidden: three linearly ordered points and a fourth incomparable point</div></div></div></div></div> <p>A semiorder, defined from a utility function as above, is a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> with the following two properties:<sup id="cite_ref-FOOTNOTEBrightwell1989_3-0" class="reference"><a href="#cite_note-FOOTNOTEBrightwell1989-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Whenever two disjoint pairs of elements are comparable, for instance 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 w<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w<x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cad47dc89753ef567d07e7725e8c31ffac5a0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.092ex; height:1.843ex;" alt="{\displaystyle w<x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fe72701bcf411046e7d2431f0cb5850ec523de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle y<z}"></span>, there must be an additional comparison among these elements, because <span class="mwe-math-element"><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(w)\leq u(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(w)\leq u(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9b4e9b16da6dae60eb19242049941674eb644a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.196ex; height:2.843ex;" alt="{\displaystyle u(w)\leq u(y)}"></span> would imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w<z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo><</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w<z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d2a8fda581213fc9281aeabd8eac0457406497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.851ex; height:1.843ex;" alt="{\displaystyle w<z}"></span> while <span class="mwe-math-element"><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(w)\geq u(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(w)\geq u(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbc458b8e6ecf6d859978c2a9e917c7b807f0f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.196ex; height:2.843ex;" alt="{\displaystyle u(w)\geq u(y)}"></span> would imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y<x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4052851a7a0fd170365775f6fc8ad079d274ee55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle y<x}"></span>. Therefore, it is impossible to have two mutually incomparable two-point <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linear orders</a>.<sup id="cite_ref-FOOTNOTEBrightwell1989_3-1" class="reference"><a href="#cite_note-FOOTNOTEBrightwell1989-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>If three elements form a linear ordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w<x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w<x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b30a7fe36848f2c7dc92b93947f650198f1cbcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.346ex; height:2.176ex;" alt="{\displaystyle w<x<y}"></span>, then every fourth point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> must be comparable to at least one of them, because <span class="mwe-math-element"><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(z)\leq u(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(z)\leq u(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0dfe3d0e4c8269f60d34ac392bb4790c949a1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.794ex; height:2.843ex;" alt="{\displaystyle u(z)\leq u(x)}"></span> would imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db38c33f2696c56320eabc39cd6d6a5d4ffc082e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle z<y}"></span> while <span class="mwe-math-element"><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(z)\geq u(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(z)\geq u(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60684c2b780fef1f2ce0cd0fa42ff3155eb0846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.794ex; height:2.843ex;" alt="{\displaystyle u(z)\geq u(x)}"></span> would imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w<z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo><</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w<z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d2a8fda581213fc9281aeabd8eac0457406497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.851ex; height:1.843ex;" alt="{\displaystyle w<z}"></span>, in either case showing that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> is comparable to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> or to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. So it is impossible to have a three-point linear order with a fourth incomparable point.<sup id="cite_ref-FOOTNOTEBrightwell1989_3-2" class="reference"><a href="#cite_note-FOOTNOTEBrightwell1989-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul> <p>Conversely, every finite partial order that avoids the two forbidden four-point orderings described above can be given utility values making it into a semiorder. <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> Therefore, rather than being a consequence of a definition in terms of utility, these forbidden orderings, or equivalent systems of <a href="/wiki/Axiom" title="Axiom">axioms</a>, can be taken as a combinatorial definition of semiorders.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> If a semiorder on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> elements is given only in terms of the order relation between its pairs of elements, obeying these axioms, then it is possible to construct a utility function that represents the order in time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}"></span>, where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> is an instance of <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>.<sup id="cite_ref-FOOTNOTEAvery1992_6-0" class="reference"><a href="#cite_note-FOOTNOTEAvery1992-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>For orderings on infinite sets of elements, the orderings that can be defined by utility functions and the orderings that can be defined by forbidden four-point orders differ from each other. For instance, if a semiorder <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo><</mo> <mo stretchy="false">)</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/a06a913707358013e41f176121fbfc0c1e3c78ef" 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,<)}"></span> (as defined by forbidden orders) includes an <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> <a href="/wiki/Total_order" title="Total order">totally ordered subset</a> then there do not exist sufficiently many sufficiently well-spaced real-numbers for it to be representable by a utility function. <a href="#CITEREFFishburn1973">Fishburn (1973)</a> supplies a precise characterization of the semiorders that may be defined numerically.<sup id="cite_ref-FOOTNOTEFishburn1973_7-0" class="reference"><a href="#cite_note-FOOTNOTEFishburn1973-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_kinds_of_order">Relation to other kinds of order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=3" title="Edit section: Relation to other kinds of order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Partial_orders">Partial orders</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=4" title="Edit section: Partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may define a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</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> from a semiorder <span class="mwe-math-element"><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"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo><</mo> <mo stretchy="false">)</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/a06a913707358013e41f176121fbfc0c1e3c78ef" 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,<)}"></span> by declaring that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> whenever either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span> 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 x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>. Of the axioms that a partial order is required to obey, reflexivity (<span class="mwe-math-element"><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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a5929024fa38cfa940f872cbaba86a7a3b2097" 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 x\leq x}"></span>) follows automatically from this definition. Antisymmetry (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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de6a6e4f44d9dfcbfaadbdcf388d4b8a6fed109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle y\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=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>) follows from the first semiorder axiom. Transitivity (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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≤</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d07572d94c207f03149ba6ee1bdc1d4b6f8b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.343ex;" alt="{\displaystyle y\leq z}"></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\leq z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d407deef131cba48a58513132e69a4db6ca2fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle x\leq z}"></span>) follows from the second semiorder axiom. Therefore, the binary relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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> defined in this way meets the three requirements of a partial order that it be reflexive, antisymmetric, and transitive. </p><p>Conversely, suppose 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 )}"> <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 partial order that has been constructed in this way from a semiorder. Then the semiorder may be recovered by declaring 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<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51b711ca7f932963cdb268b0817dc72d6258733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.676ex;" alt="{\displaystyle x\neq y}"></span>. Not every partial order leads to a semiorder in this way, however: The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, but the others do not. A partial order that includes four elements forming two two-element chains would lead to a relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,<)}"> <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,<)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06a913707358013e41f176121fbfc0c1e3c78ef" 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,<)}"></span> that violates the second semiorder axiom, and a partial order that includes four elements forming a three-element chain and an unrelated item would violate the third semiorder axiom (cf. pictures in section <a href="#Axiomatics">#Axiomatics</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Weak_orders">Weak orders</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=5" title="Edit section: Weak orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every strict weak ordering < is also a semi-order. More particularly, transitivity of < and transitivity of incomparability with respect to < together imply the above axiom 2, while transitivity of incomparability alone implies axiom 3. The semiorder shown in the top image is not a strict weak ordering, since the rightmost vertex is incomparable to its two closest left neighbors, but they are comparable. </p> <div class="mw-heading mw-heading3"><h3 id="Interval_orders">Interval orders</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=6" title="Edit section: Interval orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The semiorder defined from a utility function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> may equivalently be defined as the <a href="/wiki/Interval_order" title="Interval order">interval order</a> defined by the intervals <span class="mwe-math-element"><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(x),u(x)+1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [u(x),u(x)+1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb7db93cec5fb166a8993b4ce62d573ebaef1a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.268ex; height:2.843ex;" alt="{\displaystyle [u(x),u(x)+1]}"></span>,<sup id="cite_ref-FOOTNOTEFishburn1970_8-0" class="reference"><a href="#cite_note-FOOTNOTEFishburn1970-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> so every semiorder is an example of an interval order. A relation is a semiorder if, and only if, it can be obtained as an <a href="/wiki/Interval_order" title="Interval order">interval order</a> of unit length intervals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ell _{i},\ell _{i}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ell _{i},\ell _{i}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4347eb88644574d16b07f503c5203ad3710da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.385ex; height:2.843ex;" alt="{\displaystyle (\ell _{i},\ell _{i}+1)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Quasitransitive_relations">Quasitransitive relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=7" title="Edit section: Quasitransitive relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Amartya_K._Sen" class="mw-redirect" title="Amartya K. Sen">Amartya K. Sen</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> semi-orders were examined by <a href="/wiki/Dean_T._Jamison" class="mw-redirect" title="Dean T. Jamison">Dean T. Jamison</a> and <a href="/wiki/Lawrence_J._Lau" class="mw-redirect" title="Lawrence J. Lau">Lawrence J. Lau</a><sup id="cite_ref-FOOTNOTEJamisonLau1970_10-0" class="reference"><a href="#cite_note-FOOTNOTEJamisonLau1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and found to be a special case of <a href="/wiki/Quasitransitive_relation" title="Quasitransitive relation">quasitransitive relations</a>. In fact, every semiorder is quasitransitive,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and quasitransitivity is invariant to adding all pairs of incomparable items.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Removing all non-vertical red lines from the topmost image results in a Hasse diagram for a relation that is still quasitransitive, but violates both axiom 2 and 3; this relation might no longer be useful as a preference ordering. </p> <div class="mw-heading mw-heading2"><h2 id="Combinatorial_enumeration">Combinatorial enumeration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=8" title="Edit section: Combinatorial enumeration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number of distinct semiorders on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> unlabeled items is given by the <a href="/wiki/Catalan_number" title="Catalan number">Catalan numbers</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <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 {\frac {1}{n+1}}{\binom {2n}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n+1}}{\binom {2n}{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10504a042d5a3c56f2fb454d1daa59a9ead0becf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.859ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{n+1}}{\binom {2n}{n}},}"></span> while the number of semiorders on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> labeled items is given by the sequence<sup id="cite_ref-FOOTNOTEChandonLemairePouget1978_14-0" class="reference"><a href="#cite_note-FOOTNOTEChandonLemairePouget1978-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.6em;">1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, ... (sequence <span class="nowrap external"><a href="//oeis.org/A006531" class="extiw" title="oeis:A006531">A006531</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</div> <div class="mw-heading mw-heading2"><h2 id="Other_results">Other results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=9" title="Edit section: Other results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any finite semiorder has <a href="/wiki/Order_dimension" title="Order dimension">order dimension</a> at most three.<sup id="cite_ref-FOOTNOTERabinovitch1978_15-0" class="reference"><a href="#cite_note-FOOTNOTERabinovitch1978-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of <a href="/wiki/Linear_extension" title="Linear extension">linear extensions</a> are semiorders.<sup id="cite_ref-FOOTNOTEFishburnTrotter1992_16-0" class="reference"><a href="#cite_note-FOOTNOTEFishburnTrotter1992-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Semiorders are known to obey the <a href="/wiki/1/3%E2%80%932/3_conjecture" title="1/3–2/3 conjecture">1/3–2/3 conjecture</a>: in any finite semiorder that is not a total order, there exists a pair of elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> 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}"> <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> appears earlier than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> in between 1/3 and 2/3 of the linear extensions of the semiorder.<sup id="cite_ref-FOOTNOTEBrightwell1989_3-3" class="reference"><a href="#cite_note-FOOTNOTEBrightwell1989-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The set of semiorders on an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-element set is <i>well-graded</i>: if two semiorders on the same set differ from each other by the addition or removal 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> order relations, then it is possible to find a path 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> steps from the first semiorder to the second one, in such a way that each step of the path adds or removes a single order relation and each intermediate state in the path is itself a semiorder.<sup id="cite_ref-FOOTNOTEDoignonFalmagne1997_17-0" class="reference"><a href="#cite_note-FOOTNOTEDoignonFalmagne1997-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Comparability_graph" title="Comparability graph">incomparability graphs</a> of semiorders are called <a href="/wiki/Indifference_graph" title="Indifference graph">indifference graphs</a>, and are a special case of the <a href="/wiki/Interval_graph" title="Interval graph">interval graphs</a>.<sup id="cite_ref-FOOTNOTERoberts1969_18-0" class="reference"><a href="#cite_note-FOOTNOTERoberts1969-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTELuce1956179-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTELuce1956179_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTELuce1956179_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLuce1956">Luce (1956)</a>, p. 179.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFLuce1956">Luce (1956)</a>, Theorem 3 describes a more general situation in which the threshold for comparability between two utilities is a function of the utility rather than being identically 1; however, this does not lead to a different class of orderings.</span> </li> <li id="cite_note-FOOTNOTEBrightwell1989-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBrightwell1989_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBrightwell1989_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBrightwell1989_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBrightwell1989_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBrightwell1989">Brightwell (1989)</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">This result is typically credited to <a href="#CITEREFScottSuppes1958">Scott & Suppes (1958)</a>; see, e.g., <a href="#CITEREFRabinovitch1977">Rabinovitch (1977)</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFLuce1956">Luce (1956</a>, p. 181) used four axioms, the first two of which combine <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">asymmetry</a> and the definition of incomparability, while each of the remaining two is equivalent to one of the above prohibition properties.</span> </li> <li id="cite_note-FOOTNOTEAvery1992-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAvery1992_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAvery1992">Avery (1992)</a>.</span> </li> <li id="cite_note-FOOTNOTEFishburn1973-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFishburn1973_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFishburn1973">Fishburn (1973)</a>.</span> </li> <li id="cite_note-FOOTNOTEFishburn1970-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFishburn1970_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFishburn1970">Fishburn (1970)</a>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFSen1971">Sen (1971</a>, Section 10, p. 314) Since Luce modelled indifference between <i>x</i> and <i>y</i> as "neither <i>xRy</i> nor <i>yRx</i>", while Sen modelled it as "both <i>xRy</i> and <i>yRx</i>", Sen's remark on p.314 is likely to mean the latter property.</span> </li> <li id="cite_note-FOOTNOTEJamisonLau1970-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJamisonLau1970_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJamisonLau1970">Jamison & Lau (1970)</a>.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">since it is transitive</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">more general, to adding <i>any</i> symmetric relation</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFDeanKeller1968">Dean & Keller (1968)</a>; <a href="#CITEREFKimRoush1978">Kim & Roush (1978)</a></span> </li> <li id="cite_note-FOOTNOTEChandonLemairePouget1978-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEChandonLemairePouget1978_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFChandonLemairePouget1978">Chandon, Lemaire & Pouget (1978)</a>.</span> </li> <li id="cite_note-FOOTNOTERabinovitch1978-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERabinovitch1978_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRabinovitch1978">Rabinovitch (1978)</a>.</span> </li> <li id="cite_note-FOOTNOTEFishburnTrotter1992-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFishburnTrotter1992_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFishburnTrotter1992">Fishburn & Trotter (1992)</a>.</span> </li> <li id="cite_note-FOOTNOTEDoignonFalmagne1997-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDoignonFalmagne1997_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDoignonFalmagne1997">Doignon & Falmagne (1997)</a>.</span> </li> <li id="cite_note-FOOTNOTERoberts1969-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERoberts1969_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRoberts1969">Roberts (1969)</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output 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(1978), "Enumeration of isomorphism classes of semiorders", <i>Journal of Combinatorics, Information &System Sciences</i>, <b>3</b> (2): 58–61, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0538212">0538212</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorics%2C+Information+%26System+Sciences&rft.atitle=Enumeration+of+isomorphism+classes+of+semiorders&rft.volume=3&rft.issue=2&rft.pages=58-61&rft.date=1978&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D538212%23id-name%3DMR&rft.aulast=Kim&rft.aufirst=K.+H.&rft.au=Roush%2C+F.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuce1956" class="citation cs2"><a href="/wiki/R._Duncan_Luce" title="R. 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Duncan</a> (1956), <a rel="nofollow" class="external text" href="https://www.imbs.uci.edu/files/personnel/luce/pre1990/1956/Luce_Econometrica_1956.pdf">"Semiorders and a theory of utility discrimination"</a> <span class="cs1-format">(PDF)</span>, <i>Econometrica</i>, <b>24</b> (2): 178–191, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1905751">10.2307/1905751</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1905751">1905751</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0078632">0078632</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Econometrica&rft.atitle=Semiorders+and+a+theory+of+utility+discrimination&rft.volume=24&rft.issue=2&rft.pages=178-191&rft.date=1956&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0078632%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1905751%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1905751&rft.aulast=Luce&rft.aufirst=R.+Duncan&rft_id=https%3A%2F%2Fwww.imbs.uci.edu%2Ffiles%2Fpersonnel%2Fluce%2Fpre1990%2F1956%2FLuce_Econometrica_1956.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRabinovitch1977" class="citation cs2">Rabinovitch, Issie (1977), "The Scott-Suppes theorem on semiorders", <i>Journal of Mathematical Psychology</i>, <b>15</b> (2): 209–212, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-2496%2877%2990030-x">10.1016/0022-2496(77)90030-x</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0437404">0437404</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Psychology&rft.atitle=The+Scott-Suppes+theorem+on+semiorders&rft.volume=15&rft.issue=2&rft.pages=209-212&rft.date=1977&rft_id=info%3Adoi%2F10.1016%2F0022-2496%2877%2990030-x&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0437404%23id-name%3DMR&rft.aulast=Rabinovitch&rft.aufirst=Issie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRabinovitch1978" class="citation cs2">Rabinovitch, Issie (1978), "The dimension of semiorders", <i>Journal of Combinatorial Theory, Series A</i>, <b>25</b> (1): 50–61, <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%2F0097-3165%2878%2990030-4">10.1016/0097-3165(78)90030-4</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0498294">0498294</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory%2C+Series+A&rft.atitle=The+dimension+of+semiorders&rft.volume=25&rft.issue=1&rft.pages=50-61&rft.date=1978&rft_id=info%3Adoi%2F10.1016%2F0097-3165%2878%2990030-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0498294%23id-name%3DMR&rft.aulast=Rabinovitch&rft.aufirst=Issie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoberts1969" class="citation cs2"><a href="/wiki/Fred_S._Roberts" title="Fred S. Roberts">Roberts, Fred S.</a> (1969), "Indifference graphs", <i>Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968)</i>, Academic Press, New York, pp. 139–146, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0252267">0252267</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Indifference+graphs&rft.btitle=Proof+Techniques+in+Graph+Theory+%28Proc.+Second+Ann+Arbor+Graph+Theory+Conf.%2C+Ann+Arbor%2C+Mich.%2C+1968%29&rft.pages=139-146&rft.pub=Academic+Press%2C+New+York&rft.date=1969&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0252267%23id-name%3DMR&rft.aulast=Roberts&rft.aufirst=Fred+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScottSuppes1958" class="citation cs2"><a href="/wiki/Dana_Scott" title="Dana Scott">Scott, Dana</a>; <a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick</a> (1958), "Foundational aspects of theories of measurement", <i>The Journal of Symbolic Logic</i>, <b>23</b> (2): 113–128, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2964389">10.2307/2964389</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2964389">2964389</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0115919">0115919</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Foundational+aspects+of+theories+of+measurement&rft.volume=23&rft.issue=2&rft.pages=113-128&rft.date=1958&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0115919%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2964389%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2964389&rft.aulast=Scott&rft.aufirst=Dana&rft.au=Suppes%2C+Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSen1971" class="citation cs2">Sen, Amartya K. (July 1971), <a rel="nofollow" class="external text" href="https://www.ihs.ac.at/publications/eco/visit_profs/blume/sen.pdf">"Choice Functions and Revealed Preference"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/The_Review_of_Economic_Studies" title="The Review of Economic Studies">The Review of Economic Studies</a></i>, <b>38</b> (3): 307–317, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2296384">10.2307/2296384</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2296384">2296384</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Review+of+Economic+Studies&rft.atitle=Choice+Functions+and+Revealed+Preference&rft.volume=38&rft.issue=3&rft.pages=307-317&rft.date=1971-07&rft_id=info%3Adoi%2F10.2307%2F2296384&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2296384%23id-name%3DJSTOR&rft.aulast=Sen&rft.aufirst=Amartya+K.&rft_id=https%3A%2F%2Fwww.ihs.ac.at%2Fpublications%2Feco%2Fvisit_profs%2Fblume%2Fsen.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semiorder&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPirlotVincke1997" class="citation cs2">Pirlot, M.; Vincke, Ph. (1997), <i>Semiorders: Properties, representations, applications</i>, Theory and Decision Library. Series B: Mathematical and Statistical Methods, vol. 36, Dordrecht: Kluwer Academic Publishers Group, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-4617-3" title="Special:BookSources/0-7923-4617-3"><bdi>0-7923-4617-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1472236">1472236</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Semiorders%3A+Properties%2C+representations%2C+applications&rft.place=Dordrecht&rft.series=Theory+and+Decision+Library.+Series+B%3A+Mathematical+and+Statistical+Methods&rft.pub=Kluwer+Academic+Publishers+Group&rft.date=1997&rft.isbn=0-7923-4617-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1472236%23id-name%3DMR&rft.aulast=Pirlot&rft.aufirst=M.&rft.au=Vincke%2C+Ph.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemiorder" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_order_theory_topics" title="List of order theory topics">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a class="mw-selflink selflink">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Semiorder&oldid=1203374097">https://en.wikipedia.org/w/index.php?title=Semiorder&oldid=1203374097</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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