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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orders_with_further_algebraic_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Orders with further algebraic operations</span> </div> </a> <ul id="toc-Orders_with_further_algebraic_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orders_in_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orders_in_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Orders in algebra</span> </div> </a> <ul id="toc-Orders_in_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Functions_between_partial_orders" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Functions_between_partial_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Functions between partial orders</span> </div> </a> <ul id="toc-Functions_between_partial_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Completions_and_free_constructions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Completions_and_free_constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Completions and free constructions</span> </div> </a> <ul id="toc-Completions_and_free_constructions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Domain_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Domain_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Domain theory</span> </div> </a> <ul id="toc-Domain_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orders_in_mathematical_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orders_in_mathematical_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Orders in mathematical logic</span> </div> </a> <ul id="toc-Orders_in_mathematical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orders_in_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orders_in_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Orders in topology</span> </div> </a> <ul id="toc-Orders_in_topology-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p> <a href="/wiki/Order_theory" title="Order theory">Order theory</a> is a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> that studies various kinds of objects (often <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. </p><p>An alphabetical list of many notions of order theory can be found in the <a href="/wiki/Order_theory_glossary" class="mw-redirect" title="Order theory glossary">order theory glossary</a>. See also <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a>, <a href="/wiki/Extreme_value" class="mw-redirect" title="Extreme value">extreme value</a> and <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partially ordered set</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">Totally ordered set</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total preorder</a></li> <li><a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">Chain</a></li> <li><a href="/wiki/Trichotomy_(mathematics)" class="mw-redirect" title="Trichotomy (mathematics)">Trichotomy</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real number line</a></li></ul></li> <li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">Strict order</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> <ul><li><a href="/wiki/Directed_acyclic_graph" title="Directed acyclic graph">Directed acyclic graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality (order theory)</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Distinguished_elements_of_partial_orders">Distinguished elements of partial orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=2" title="Edit section: Distinguished elements of partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Greatest_element" class="mw-redirect" title="Greatest element">Greatest element</a> (maximum, top, unit), <a href="/wiki/Least_element" class="mw-redirect" title="Least element">Least element</a> (minimum, bottom, zero)</li> <li><a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">Maximal element</a>, <a href="/wiki/Minimal_element" class="mw-redirect" title="Minimal element">minimal element</a></li> <li><a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">Upper bound</a> <ul><li><a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">Least upper bound</a> (supremum, join)</li> <li><a href="/wiki/Greatest_lower_bound" class="mw-redirect" title="Greatest lower bound">Greatest lower bound</a> (infimum, meet)</li> <li><a href="/wiki/Limit_superior_and_limit_inferior" class="mw-redirect" title="Limit superior and limit inferior">Limit superior and limit inferior</a></li></ul></li> <li><a href="/w/index.php?title=Irreducible_element_(order_theory)&action=edit&redlink=1" class="new" title="Irreducible element (order theory) (page does not exist)">Irreducible element</a></li> <li><a href="/w/index.php?title=Prime_element_(order_theory)&action=edit&redlink=1" class="new" title="Prime element (order theory) (page does not exist)">Prime element</a></li> <li><a href="/wiki/Compact_element" title="Compact element">Compact element</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Subsets_of_partial_orders">Subsets of partial orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=3" title="Edit section: Subsets of partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a> and <a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">coinitial</a> set, sometimes also called <a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">dense</a></li> <li><a href="/w/index.php?title=Meet-dense_set&action=edit&redlink=1" class="new" title="Meet-dense set (page does not exist)">Meet-dense set</a> and <a href="/w/index.php?title=Join-dense_set&action=edit&redlink=1" class="new" title="Join-dense set (page does not exist)">join-dense set</a></li> <li><a href="/wiki/Linked_set" title="Linked set">Linked set</a> (upwards and downwards)</li> <li><a href="/wiki/Directed_set" title="Directed set">Directed set</a> (upwards and downwards)</li> <li><a href="/wiki/Centered_set" title="Centered set">centered</a> and <a href="/wiki/Centered_set" title="Centered set">σ-centered set</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net (mathematics)</a></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a> and lower set</li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a> and <a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filter</a> <ul><li><a href="/wiki/Ultrafilter" title="Ultrafilter">Ultrafilter</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Special_types_of_partial_orders">Special types of partial orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=4" title="Edit section: Special types of partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness (order theory)</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense order</a></li> <li><a href="/wiki/Distributivity_(order_theory)" title="Distributivity (order theory)">Distributivity (order theory)</a> <ul><li><a href="/wiki/Modular_lattice" title="Modular lattice">modular lattice</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a></li> <li><a href="/wiki/Completely_distributive_lattice" title="Completely distributive lattice">completely distributive lattice</a></li></ul></li> <li><a href="/wiki/Ascending_chain_condition" title="Ascending chain condition">Ascending chain condition</a> <ul><li><a href="/wiki/Infinite_descending_chain" class="mw-redirect" title="Infinite descending chain">Infinite descending chain</a></li></ul></li> <li><a href="/wiki/Countable_chain_condition" title="Countable chain condition">Countable chain condition</a>, often abbreviated as <i>ccc</i></li> <li><a href="/wiki/Knaster%27s_condition" title="Knaster's condition">Knaster's condition</a>, sometimes denoted <i>property (K)</i></li></ul> <div class="mw-heading mw-heading3"><h3 id="Well-orders"><a href="/wiki/Well-order" title="Well-order">Well-orders</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=5" title="Edit section: Well-orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded relation</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Completeness_properties"><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness properties</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=6" title="Edit section: Completeness properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li>(Directed) <a href="/wiki/Complete_partial_order" title="Complete partial order">complete partial order</a>, (d)cpo</li> <li><a href="/wiki/Bounded_complete" class="mw-redirect" title="Bounded complete">Bounded complete</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete lattice</a> <ul><li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li></ul></li> <li><a href="/wiki/Infinite_divisibility" title="Infinite divisibility">Infinite divisibility</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Orders_with_further_algebraic_operations">Orders with further <a href="/wiki/Abstract_algebra" title="Abstract algebra">algebraic</a> operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=7" title="Edit section: Orders with further algebraic operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a> <ul><li><a href="/wiki/Relatively_complemented_lattice" class="mw-redirect" title="Relatively complemented lattice">Relatively complemented lattice</a></li></ul></li> <li><a href="/wiki/Complete_Heyting_algebra" title="Complete Heyting algebra">Complete Heyting algebra</a> <ul><li><a href="/wiki/Pointless_topology" title="Pointless topology">Pointless topology</a></li></ul></li> <li><a href="/wiki/MV-algebra" title="MV-algebra">MV-algebra</a></li> <li><a href="/wiki/Ockham_algebra" title="Ockham algebra">Ockham algebras</a>: <ul><li><a href="/wiki/Stone_algebra" title="Stone algebra">Stone algebra</a></li> <li><a href="/wiki/De_Morgan_algebra" title="De Morgan algebra">De Morgan algebra</a> <ul><li><a href="/wiki/Kleene_algebra_(with_involution)" class="mw-redirect" title="Kleene algebra (with involution)">Kleene algebra (with involution)</a></li> <li><a href="/wiki/%C5%81ukasiewicz%E2%80%93Moisil_algebra" title="Łukasiewicz–Moisil algebra">Łukasiewicz–Moisil algebra</a></li></ul></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra (structure)</a> <ul><li><a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a></li> <li><a href="/wiki/Complete_Boolean_algebra" title="Complete Boolean algebra">Complete Boolean algebra</a></li></ul></li></ul></li> <li><a href="/wiki/Orthocomplemented_lattice" class="mw-redirect" title="Orthocomplemented lattice">Orthocomplemented lattice</a></li> <li><a href="/wiki/Quantale" title="Quantale">Quantale</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Orders_in_algebra">Orders in <a href="/wiki/Algebra" title="Algebra">algebra</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=8" title="Edit section: Orders in algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Partially_ordered_monoid" class="mw-redirect" title="Partially ordered monoid">Partially ordered monoid</a></li> <li><a href="/wiki/Ordered_group" class="mw-redirect" title="Ordered group">Ordered group</a> <ul><li><a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a></li></ul></li> <li><a href="/wiki/Ordered_ring" title="Ordered ring">Ordered ring</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a></li> <li><a href="/wiki/Artinian_ring" title="Artinian ring">Artinian ring</a></li> <li><a href="/wiki/Noetherian" title="Noetherian">Noetherian</a></li> <li><a href="/wiki/Linearly_ordered_group" title="Linearly ordered group">Linearly ordered group</a></li> <li><a href="/wiki/Monomial_order" title="Monomial order">Monomial order</a></li> <li><a href="/wiki/Weak_order_of_permutations" class="mw-redirect" title="Weak order of permutations">Weak order of permutations</a></li> <li><a href="/wiki/Bruhat_order" title="Bruhat order">Bruhat order</a> on a Coxeter group</li> <li><a href="/wiki/Incidence_algebra" title="Incidence algebra">Incidence algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Functions_between_partial_orders"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a> between partial orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=9" title="Edit section: Functions between partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Monotonic" class="mw-redirect" title="Monotonic">Monotonic</a></li> <li><a href="/wiki/Pointwise_order" class="mw-redirect" title="Pointwise order">Pointwise order</a> of functions</li> <li><a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a></li> <li><a href="/wiki/Order_embedding" title="Order embedding">Order embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Order isomorphism</a></li> <li><a href="/wiki/Closure_operator" title="Closure operator">Closure operator</a></li> <li>Functions that <a href="/wiki/Limit-preserving_function_(order_theory)" title="Limit-preserving function (order theory)">preserve</a> suprema/infima</li></ul> <div class="mw-heading mw-heading2"><h2 id="Completions_and_free_constructions"><a href="/wiki/Completion_(order_theory)" class="mw-redirect" title="Completion (order theory)">Completions</a> and <a href="/wiki/Free_object" title="Free object">free constructions</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=10" title="Edit section: Completions and free constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dedekind_completion" class="mw-redirect" title="Dedekind completion">Dedekind completion</a></li> <li><a href="/w/index.php?title=Ideal_completion&action=edit&redlink=1" class="new" title="Ideal completion (page does not exist)">Ideal completion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Domain_theory">Domain theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=11" title="Edit section: Domain theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Domain_theory" title="Domain theory">Domain theory</a></div> <ul><li><a href="/wiki/Way-below_relation" class="mw-redirect" title="Way-below relation">Way-below relation</a></li> <li><a href="/wiki/Continuous_poset" title="Continuous poset">Continuous poset</a> <ul><li><a href="/wiki/Continuous_lattice" class="mw-redirect" title="Continuous lattice">Continuous lattice</a></li></ul></li> <li><a href="/wiki/Algebraic_poset" class="mw-redirect" title="Algebraic poset">Algebraic poset</a> <ul><li><a href="/wiki/Scott_domain" title="Scott domain">Scott domain</a></li> <li><a href="/wiki/Algebraic_lattice" class="mw-redirect" title="Algebraic lattice">Algebraic lattice</a></li></ul></li> <li><a href="/wiki/Scott_information_system" title="Scott information system">Scott information system</a></li> <li><a href="/wiki/Powerdomain" class="mw-redirect" title="Powerdomain">Powerdomain</a></li> <li><a href="/wiki/Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a></li> <li><a href="/wiki/Scott_continuity" title="Scott continuity">Scott continuity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Orders_in_mathematical_logic">Orders in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=12" title="Edit section: Orders in mathematical logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lindenbaum_algebra" class="mw-redirect" title="Lindenbaum algebra">Lindenbaum algebra</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a> <ul><li><a href="/wiki/Hausdorff_maximality_theorem" class="mw-redirect" title="Hausdorff maximality theorem">Hausdorff maximality theorem</a></li></ul></li> <li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Ultrafilter" title="Ultrafilter">Ultrafilter</a></li> <li><a href="/wiki/Ultrafilter_lemma" class="mw-redirect" title="Ultrafilter lemma">Ultrafilter lemma</a></li> <li><a href="/wiki/Tree_(set_theory)" title="Tree (set theory)">Tree (set theory)</a></li> <li><a href="/wiki/Tree_(descriptive_set_theory)" title="Tree (descriptive set theory)">Tree (descriptive set theory)</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Absorption_law" title="Absorption law">Absorption law</a></li> <li><a href="/wiki/Prewellordering" title="Prewellordering">Prewellordering</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Orders_in_topology">Orders in <a href="/wiki/Topology" title="Topology">topology</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_order_theory_topics&action=edit&section=13" title="Edit section: Orders in topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stone_duality" title="Stone duality">Stone duality</a> <ul><li><a href="/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" title="Stone's representation theorem for Boolean algebras">Stone's representation theorem for Boolean algebras</a></li></ul></li> <li><a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization (pre)order</a></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a> of a total order (open interval topology)</li> <li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a></li> <li><a href="/wiki/Upper_topology" title="Upper topology">Upper topology</a></li> <li><a href="/wiki/Scott_topology" class="mw-redirect" title="Scott topology">Scott topology</a> <ul><li><a href="/wiki/Scott_continuity" title="Scott continuity">Scott continuity</a></li></ul></li> <li><a href="/wiki/Lawson_topology" title="Lawson topology">Lawson topology</a></li> <li><a href="/wiki/Finer_topology" class="mw-redirect" title="Finer topology">Finer topology</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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.navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order_theory" title="Template:Order theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" title="Special:EditPage/Template:Order theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Order_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 class="mw-selflink selflink">Topics</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li> <li><a href="/wiki/Category:Order_theory" title="Category:Order theory">Category</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a href="/wiki/Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a 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 class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" 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