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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Complete partial order</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="Go to an article in another language. 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data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%BD%D0%BE_%D1%83%D0%BF%D0%BE%D1%80%D1%8F%D0%B4%D0%BE%D1%87%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE#Полное_частично_упорядоченное_множество" title="Частично упорядоченное множество – Russian" lang="ru" hreflang="ru" data-title="Частично упорядоченное множество" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%8C%E5%85%A8%E5%81%8F%E5%BA%8F" title="完全偏序 – Chinese" lang="zh" hreflang="zh" data-title="完全偏序" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical phrase</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the phrase <b>complete partial order</b> is variously used to refer to at least three similar, but distinct, classes of <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a>, characterized by particular <a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completeness properties</a>. Complete partial orders play a central role in <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>: in <a href="/wiki/Denotational_semantics" title="Denotational semantics">denotational semantics</a> and <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term <b>complete partial order</b>, abbreviated <b>cpo</b>, has several possible meanings depending on context. </p><p>A partially ordered set is a <b>directed-complete partial order</b> (<b>dcpo</b>) if each of its <a href="/wiki/Directed_set" title="Directed set">directed subsets</a> has a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>. (A subset of a partial order is directed if it is <a href="/wiki/Empty_set" title="Empty set">non-empty</a> and every pair of elements has an upper bound in the subset.) In the literature, dcpos sometimes also appear under the label <b>up-complete poset</b>. </p><p>A <b>pointed directed-complete partial order</b> (<b>pointed dcpo</b>, sometimes abbreviated <b>cppo</b>), is a dcpo with a <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a> (usually denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"></span>). Formulated differently, a pointed dcpo has a supremum for every directed <i>or empty</i> subset. The term <b>chain-complete partial order</b> is also used, because of the characterization of pointed dcpos as posets in which every <a href="/wiki/Chain_(ordered_set)" class="mw-redirect" title="Chain (ordered set)">chain</a> has a supremum. </p><p>A related notion is that of <b>ω-complete partial order</b> (<b>ω-cpo</b>). These are posets in which every ω-chain (<span class="mwe-math-element"><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_{1}\leq x_{2}\leq x_{3}\leq ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≤</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\leq x_{2}\leq x_{3}\leq ...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843f64870b70a4914510ed0f10ec9076425695dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.517ex; height:2.343ex;" alt="{\displaystyle x_{1}\leq x_{2}\leq x_{3}\leq ...}"></span>) has a supremum that belongs to the poset. The same notion can be extended to other <a href="/wiki/Cardinality" title="Cardinality">cardinalities</a> of chains. <sup id="cite_ref-markowsky-1976_1-0" class="reference"><a href="#cite_note-markowsky-1976-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Every dcpo is an ω-cpo, since every ω-chain is a directed set, but the <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a> is not true. However, every ω-cpo with a <a href="/wiki/Domain_theory#Bases_of_domains" title="Domain theory">basis</a> is also a dcpo (with the same basis).<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> An ω-cpo (dcpo) with a basis is also called a <b>continuous</b> ω-cpo (or continuous dcpo). </p><p>Note that <i>complete partial order</i> is never used to mean a poset in which <i>all</i> subsets have suprema; the terminology <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> is used for this concept. </p><p>Requiring the existence of directed suprema can be motivated by viewing directed sets as generalized approximation sequences and suprema as <i>limits</i> of the respective (approximative) computations. This intuition, in the context of denotational semantics, was the motivation behind the development of <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>. </p><p>The <a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">dual</a> notion of a directed-complete partial order is called a <b>filtered-complete partial order</b>. However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly. </p><p>By analogy with the <a href="/wiki/Dedekind%E2%80%93MacNeille_completion" title="Dedekind–MacNeille completion">Dedekind–MacNeille completion</a> of a partially ordered set, every partially ordered set can be extended uniquely to a minimal dcpo.<sup id="cite_ref-markowsky-1976_1-1" class="reference"><a href="#cite_note-markowsky-1976-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Every finite poset is directed complete.</li> <li>All <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattices</a> are also directed complete.</li> <li>For any poset, the set of all non-empty <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filters</a>, ordered by <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">subset inclusion</a>, is a dcpo. Together with the empty filter it is also pointed. If the order has binary <a href="/wiki/Join_and_meet" title="Join and meet">meets</a>, then this construction (including the empty filter) actually yields a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a>.</li> <li>Every set <i>S</i> can be turned into a pointed dcpo by adding a least element ⊥ and introducing a flat order with ⊥ ≤ <i>s</i> and s ≤ <i>s</i> for every <i>s</i> in <i>S</i> and no other order relations.</li> <li>The set of all <a href="/wiki/Partial_function" title="Partial function">partial functions</a> on some given set <i>S</i> can be ordered by defining <i>f</i> ≤ <i>g</i> if and only if <i>g</i> extends <i>f</i>, i.e. if the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of <i>f</i> is a subset of the domain of <i>g</i> and the values of <i>f</i> and <i>g</i> agree on all inputs for which they are both defined. (Equivalently, <i>f</i> ≤ <i>g</i> if and only if <i>f</i> ⊆ <i>g</i> where <i>f</i> and <i>g</i> are identified with their respective <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a>.) This order is a pointed dcpo, where the least element is the nowhere-defined partial function (with empty domain). In fact, ≤ is also <a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">bounded complete</a>. This example also demonstrates why it is not always natural to have a greatest element.</li> <li>The set of all <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> <a href="/wiki/Subset" title="Subset">subsets</a> of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <i>V</i>, ordered by <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusion</a>.</li> <li>The set of all partial <a href="/wiki/Choice_function" title="Choice function">choice functions</a> on a collection of <a href="/wiki/Empty_set" title="Empty set">non-empty</a> sets, ordered by restriction.</li> <li>The set of all <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, ordered by inclusion.</li> <li>The <a href="/wiki/Specialization_order" class="mw-redirect" title="Specialization order">specialization order</a> of any <a href="/wiki/Sober_space" title="Sober space">sober space</a> is a dcpo.</li> <li>Let us use the term “<a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a>” as a set of <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentences</a> closed under consequence (for defining notion of consequence, let us use e.g. <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>'s algebraic approach<sup id="cite_ref-Tar-BizIg_3-0" class="reference"><a href="#cite_note-Tar-BizIg-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BurSan-UnivAlg_4-0" class="reference"><a href="#cite_note-BurSan-UnivAlg-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>). There are interesting theorems that concern a set of deductive systems being a directed-complete partial ordering.<sup id="cite_ref-seqdcpo_5-0" class="reference"><a href="#cite_note-seqdcpo-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Also, a set of deductive systems can be chosen to have a least element in a natural way (so that it can be also a pointed dcpo), because the set of all consequences of the empty set (i.e. “the set of the logically provable/logically valid sentences”) is (1) a deductive system (2) contained by all deductive systems.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Characterizations">Characterizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=3" title="Edit section: Characterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An ordered set is a dcpo if and only if every non-empty <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> has a supremum. As a corollary, an ordered set is a pointed dcpo if and only if every (possibly empty) chain has a supremum, i.e., if and only if it is <a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">chain-complete</a>.<sup id="cite_ref-markowsky-1976_1-2" class="reference"><a href="#cite_note-markowsky-1976-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Proofs rely on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. </p><p>Alternatively, an ordered set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is a pointed dcpo if and only if every <a href="/wiki/Order-preserving" class="mw-redirect" title="Order-preserving">order-preserving</a> self-map 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> has a least <a href="/wiki/Fixpoint" class="mw-redirect" title="Fixpoint">fixpoint</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Continuous_functions_and_fixed-points">Continuous functions and fixed-points</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=4" title="Edit section: Continuous functions and fixed-points"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i> between two dcpos <i>P</i> and <i>Q</i> is called <b><a href="/wiki/Scott_continuity" title="Scott continuity">(Scott) continuous</a></b> if it maps directed sets to directed sets while preserving their suprema: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(D)\subseteq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(D)\subseteq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ab4f45b83589b438d4b84d3ec8f94362bfd162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.949ex; height:2.843ex;" alt="{\displaystyle f(D)\subseteq Q}"></span> is directed for every directed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subseteq P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊆</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subseteq P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5d24ab7cff6c595b78324a40ad01c72659f8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.768ex; height:2.343ex;" alt="{\displaystyle D\subseteq P}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\sup D)=\sup f(D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">sup</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\sup D)=\sup f(D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/948be5fd82585d34fbe085725372e4cf62a163be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.9ex; height:2.843ex;" alt="{\displaystyle f(\sup D)=\sup f(D)}"></span> for every directed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subseteq P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊆</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subseteq P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5d24ab7cff6c595b78324a40ad01c72659f8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.768ex; height:2.343ex;" alt="{\displaystyle D\subseteq P}"></span>.</li></ul> <p>Note that every continuous function between dcpos is a <a href="/wiki/Monotone_function#Monotonicity_in_order_theory" class="mw-redirect" title="Monotone function">monotone function</a>. This notion of continuity is equivalent to the <a href="/wiki/Topological_continuity" class="mw-redirect" title="Topological continuity">topological continuity</a> induced by the <a href="/wiki/Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a>. </p><p>The set of all continuous functions between two dcpos <i>P</i> and <i>Q</i> is denoted [<i>P</i> → <i>Q</i>]. Equipped with the <a href="/wiki/Pointwise#Pointwise_relations" title="Pointwise">pointwise order</a>, this is again a dcpo, and pointed whenever <i>Q</i> is pointed. Thus the complete partial orders with Scott-continuous maps form a <a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">cartesian closed</a> <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>.<sup id="cite_ref-barendregt1984_9-0" class="reference"><a href="#cite_note-barendregt1984-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Every order-preserving self-map <i>f</i> of a pointed dcpo (<i>P</i>, ⊥) has a least fixed-point.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> If <i>f</i> is continuous then this fixed-point is equal to the supremum of the <a href="/wiki/Iterated_function" title="Iterated function">iterates</a> (⊥, <i>f</i> (⊥), <i>f</i> (<i>f</i> (⊥)), … <i>f</i><sup> <i>n</i></sup>(⊥), …) of ⊥ (see also the <a href="/wiki/Kleene_fixed-point_theorem" title="Kleene fixed-point theorem">Kleene fixed-point theorem</a>). </p><p>Another fixed point theorem is the <a href="/wiki/Bourbaki-Witt_theorem" class="mw-redirect" title="Bourbaki-Witt theorem">Bourbaki-Witt theorem</a>, stating that 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is a function from a dcpo to itself with the property 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 f(x)\geq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\geq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7b27daf9286032e04f9c26d8021405066d4ff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)\geq x}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>, 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> has a fixed point. This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.<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><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> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">Algebraic posets</a></li> <li><a href="/wiki/Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complete_partial_order&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-markowsky-1976-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-markowsky-1976_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-markowsky-1976_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-markowsky-1976_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMarkowsky1976" class="citation cs2">Markowsky, George (1976), "Chain-complete posets and directed sets with applications", <i>Algebra Universalis</i>, <b>6</b> (1): 53–68, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02485815">10.1007/bf02485815</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=0398913">0398913</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16718857">16718857</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algebra+Universalis&rft.atitle=Chain-complete+posets+and+directed+sets+with+applications&rft.volume=6&rft.issue=1&rft.pages=53-68&rft.date=1976&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0398913%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16718857%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf02485815&rft.aulast=Markowsky&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramskyGabbayMaibaum1994" class="citation book cs1"><a href="/wiki/Samson_Abramsky" title="Samson Abramsky">Abramsky S</a>, <a href="/wiki/Dov_Gabbay" title="Dov Gabbay">Gabbay DM</a>, Maibaum TS (1994). <i>Handbook of Logic in Computer Science, volume 3</i>. Oxford: Clarendon Press. Prop 2.2.14, pp. 20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198537625" title="Special:BookSources/9780198537625"><bdi>9780198537625</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Logic+in+Computer+Science%2C+volume+3&rft.place=Oxford&rft.pages=Prop+2.2.14%2C+pp.+20&rft.pub=Clarendon+Press&rft.date=1994&rft.isbn=9780198537625&rft.aulast=Abramsky&rft.aufirst=S&rft.au=Gabbay%2C+DM&rft.au=Maibaum%2C+TS&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></span> </li> <li id="cite_note-Tar-BizIg-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tar-BizIg_3-0">^</a></b></span> <span class="reference-text">Tarski, Alfred: Bizonyítás és igazság / Válogatott tanulmányok. Gondolat, Budapest, 1990. (Title means: Proof and truth / Selected papers.)</span> </li> <li id="cite_note-BurSan-UnivAlg-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-BurSan-UnivAlg_4-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.uwaterloo.ca/~snburris/index.html">Stanley N. Burris</a> and H.P. Sankappanavar: <a rel="nofollow" class="external text" href="http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html">A Course in Universal Algebra</a></span> </li> <li id="cite_note-seqdcpo-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-seqdcpo_5-0">^</a></b></span> <span class="reference-text">See online in p. 24 exercises 5–6 of §5 in <a rel="nofollow" class="external autonumber" href="https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf">[1]</a>. Or, on paper, see <a href="#_note-Tar-BizIg">Tar:BizIg</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoubault-Larrecq2015" class="citation web cs1">Goubault-Larrecq, Jean (February 23, 2015). <a rel="nofollow" class="external text" href="https://topology.lmf.cnrs.fr/iwamuras-lemma-kowalskys-theorem-and-ordinals/">"Iwamura's Lemma, Markowsky's Theorem and ordinals"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">January 6,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Iwamura%27s+Lemma%2C+Markowsky%27s+Theorem+and+ordinals&rft.date=2015-02-23&rft.aulast=Goubault-Larrecq&rft.aufirst=Jean&rft_id=https%3A%2F%2Ftopology.lmf.cnrs.fr%2Fiwamuras-lemma-kowalskys-theorem-and-ordinals%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn" class="citation book cs1">Cohn, Paul Moritz. <i>Universal Algebra</i>. Harper and Row. p. 33.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Universal+Algebra&rft.pages=33&rft.pub=Harper+and+Row&rft.aulast=Cohn&rft.aufirst=Paul+Moritz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoubault-Larrecq2018" class="citation web cs1">Goubault-Larrecq, Jean (January 28, 2018). <a rel="nofollow" class="external text" href="https://topology.lmf.cnrs.fr/markowsky-or-cohn/">"Markowsky or Cohn?"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">January 6,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Markowsky+or+Cohn%3F&rft.date=2018-01-28&rft.aulast=Goubault-Larrecq&rft.aufirst=Jean&rft_id=https%3A%2F%2Ftopology.lmf.cnrs.fr%2Fmarkowsky-or-cohn%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></span> </li> <li id="cite_note-barendregt1984-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-barendregt1984_9-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Henk_Barendregt" title="Henk Barendregt">Barendregt, Henk</a>, <a rel="nofollow" class="external text" href="http://www.elsevier.com/wps/find/bookdescription.cws_home/501727/description#description"><i>The lambda calculus, its syntax and semantics</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040823174002/http://www.elsevier.com/wps/find/bookdescription.cws_home/501727/description#description">Archived</a> 2004-08-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/North-Holland_Publishing_Company" class="mw-redirect" title="North-Holland Publishing Company">North-Holland</a> (1984)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">This is a strengthening of the <a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a> sometimes referred to as "Pataraia’s theorem". For example, see Section 4.1 of <a rel="nofollow" class="external text" href="https://doi.org/10.4230/LIPIcs.TYPES.2016.6">"Realizability at Work: Separating Two Constructive Notions of Finiteness"</a> (2016) by Bezem et al. See also Chapter 4 of <i>The foundations of program verification</i> (1987), 2nd edition, Jacques Loeckx and Kurt Sieber, John Wiley & Sons, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-91282-4" title="Special:BookSources/0-471-91282-4">0-471-91282-4</a>, where the Knaster–Tarski theorem, formulated over pointed dcpo's, is given to prove as exercise 4.3-5 on page 90.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1949" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1949), "Sur le théorème de Zorn", <i>Archiv der Mathematik</i>, <b>2</b> (6): 434–437 (1951), <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02036949">10.1007/bf02036949</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=0047739">0047739</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117826806">117826806</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+der+Mathematik&rft.atitle=Sur+le+th%C3%A9or%C3%A8me+de+Zorn&rft.volume=2&rft.issue=6&rft.pages=434-437+%281951%29&rft.date=1949&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0047739%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117826806%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf02036949&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWitt1951" class="citation cs2"><a href="/wiki/Ernst_Witt" title="Ernst Witt">Witt, Ernst</a> (1951), "Beweisstudien zum Satz von M. Zorn", <i><a href="/wiki/Mathematische_Nachrichten" title="Mathematische Nachrichten">Mathematische Nachrichten</a></i>, <b>4</b>: 434–438, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmana.3210040138">10.1002/mana.3210040138</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=0039776">0039776</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Nachrichten&rft.atitle=Beweisstudien+zum+Satz+von+M.+Zorn&rft.volume=4&rft.pages=434-438&rft.date=1951&rft_id=info%3Adoi%2F10.1002%2Fmana.3210040138&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0039776%23id-name%3DMR&rft.aulast=Witt&rft.aufirst=Ernst&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span>.</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=Complete_partial_order&action=edit&section=7" title="Edit section: References"><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="CITEREFDavey,_B.A.Priestley,_H._A.2002" class="citation book cs1">Davey, B.A.; <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">Priestley, H. A.</a> (2002). <a href="/wiki/Introduction_to_Lattices_and_Order" title="Introduction to Lattices and Order"><i>Introduction to Lattices and Order</i></a> (Second ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-78451-4" title="Special:BookSources/0-521-78451-4"><bdi>0-521-78451-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Lattices+and+Order&rft.edition=Second&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=0-521-78451-4&rft.au=Davey%2C+B.A.&rft.au=Priestley%2C+H.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComplete+partial+order" class="Z3988"></span></li></ul>    </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=Complete_partial_order&oldid=1257116257">https://en.wikipedia.org/w/index.php?title=Complete_partial_order&oldid=1257116257</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Order theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 09:22<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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