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href="#Functions_between_prefix_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Functions between prefix orders</span> </div> </a> <ul id="toc-Functions_between_prefix_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Product_and_union" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Product_and_union"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Product and union</span> </div> </a> <ul id="toc-Product_and_union-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isomorphism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isomorphism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Isomorphism</span> </div> </a> <ul id="toc-Isomorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-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" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button 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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>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, especially <a href="/wiki/Order_theory" title="Order theory">order theory</a>, a <b>prefix ordered set</b> generalizes the intuitive concept of a <a href="/wiki/Tree_(set_theory)" title="Tree (set theory)">tree</a> by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a> as a set of functions from <i>time</i> (a <a href="/wiki/Totally-ordered_set" class="mw-redirect" title="Totally-ordered set">totally-ordered set</a>) to some <a href="/wiki/Phase_space" title="Phase space">phase space</a>. In this case, the elements of the set are usually referred to as <i>executions</i> of the system. </p><p>The name <i>prefix order</i> stems from the prefix order on words, which is a special kind of <a href="/wiki/Substring" title="Substring">substring</a> relation and, because of its discrete character, a tree. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prefix_order&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>prefix order</b> is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> "≤" over a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>P</i> which is <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>, <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>, and <b>downward total</b>, i.e., for all <i>a</i>, <i>b</i>, and <i>c</i> in <i>P</i>, we have that: </p> <ul><li><i>a ≤ a</i> (reflexivity);</li> <li>if <i>a ≤ b</i> and <i>b ≤ a</i> then <i>a</i> = <i>b</i> (antisymmetry);</li> <li>if <i>a ≤ b</i> and <i>b ≤ c</i> then <i>a ≤ c</i> (transitivity);</li> <li>if <i>a ≤ c</i> and <i>b ≤ c</i> then <i>a ≤ b</i> or <i>b ≤ a</i> (downward totality).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Functions_between_prefix_orders">Functions between prefix orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prefix_order&action=edit&section=2" title="Edit section: Functions between prefix orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While between partial orders it is usual to consider <a href="/wiki/Order-preserving_function" class="mw-redirect" title="Order-preserving function">order-preserving functions</a>, the most important type of functions between prefix orders are so-called <b>history preserving</b> functions. Given a prefix ordered set <i>P</i>, a <b>history</b> of a point <i>p</i>∈<i>P</i> is the (by definition totally ordered) set <i>p</i>− = {<i>q</i> | <i>q</i> ≤ <i>p</i>}. A function <i>f</i>: <i>P</i> → <i>Q</i> between prefix orders <i>P</i> and <i>Q</i> is then <b>history preserving</b> if and only if for every <i>p</i>∈<i>P</i> we find <i>f</i>(<i>p</i>−) = <i>f</i>(<i>p</i>)−. Similarly, a <b>future</b> of a point <i>p</i>∈<i>P</i> is the (prefix ordered) set <i>p</i>+ = {<i>q</i> | <i>p</i> ≤ <i>q</i>} and <i>f</i> is future preserving if for all <i>p</i>∈<i>P</i> we find <i>f</i>(<i>p</i>+) = <i>f</i>(<i>p</i>)+. </p><p>Every history preserving function and every future preserving function is also order preserving, but not vice versa. In the theory of dynamical systems, history preserving maps capture the intuition that the behavior in one system is a <i>refinement</i> of the behavior in another. Furthermore, functions that are history and future preserving <a href="/wiki/Surjective_function" title="Surjective function">surjections</a> capture the notion of <a href="/wiki/Bisimulation" title="Bisimulation">bisimulation</a> between systems, and thus the intuition that a given refinement is <i>correct</i> with respect to a specification. </p><p>The <a href="/wiki/Range_of_a_function" title="Range of a function">range</a> of a history preserving function is always a <a href="/wiki/Prefix_closed" class="mw-redirect" title="Prefix closed">prefix closed</a> subset, where a subset <i>S ⊆ P</i> is <b>prefix closed</b> if for all <i>s,t ∈ P</i> with <i>t∈S</i> and <i>s≤t</i> we find <i>s∈S</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Product_and_union">Product and union</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prefix_order&action=edit&section=3" title="Edit section: Product and union"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Taking history preserving maps as <i>morphisms</i> in the <a href="/wiki/Category_theory" title="Category theory">category</a> of prefix orders leads to a notion of product that is <i>not</i> the Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an <i>arbitrary interleaving</i> of the original prefix orders. The union of two prefix orders is the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a>, as it is with partial orders. </p> <div class="mw-heading mw-heading2"><h2 id="Isomorphism">Isomorphism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prefix_order&action=edit&section=4" title="Edit section: Isomorphism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any bijective history preserving function is an <a href="/wiki/Order_isomorphism" title="Order isomorphism">order isomorphism</a>. Furthermore, if for a given prefix ordered set <i>P</i> we construct the set <i>P- ≜ { p- | p∈ P}</i> we find that this set is prefix ordered by the subset relation ⊆, and furthermore, that the function <i>max: P- → P</i> is an isomorphism, where <i>max(S)</i> returns for each set <i>S∈P-</i> the maximum element in terms of the order on P (i.e. <i>max(p-) ≜ p</i>). </p> <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=Prefix_order&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCuijpers2013" class="citation news cs1">Cuijpers, Pieter (2013). <a rel="nofollow" class="external text" href="http://www.dcm-workshop.org.uk/2013/dcm-2013-pre-proc.pdf#page=30">"Prefix Orders as a General Model of Dynamics"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the 9th International Workshop on Developments in Computational Models (DCM)</i>. pp. 25–29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+9th+International+Workshop+on+Developments+in+Computational+Models+%28DCM%29&rft.atitle=Prefix+Orders+as+a+General+Model+of+Dynamics&rft.pages=25-29&rft.date=2013&rft.aulast=Cuijpers&rft.aufirst=Pieter&rft_id=http%3A%2F%2Fwww.dcm-workshop.org.uk%2F2013%2Fdcm-2013-pre-proc.pdf%23page%3D30&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrefix+order" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCuijpers2013" class="citation journal cs1">Cuijpers, Pieter (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.4204%2FEPTCS.120.7">"The Categorical Limit of a Sequence of Dynamical Systems"</a>. <i>EPTCS 120: Proceedings EXPRESS/SOS 2013</i>. <b>120</b>: 78–92. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1307.7445">1307.7445</a></span>. <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/10.4204%2FEPTCS.120.7">10.4204/EPTCS.120.7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=EPTCS+120%3A+Proceedings+EXPRESS%2FSOS+2013&rft.atitle=The+Categorical+Limit+of+a+Sequence+of+Dynamical+Systems&rft.volume=120&rft.pages=78-92&rft.date=2013&rft_id=info%3Aarxiv%2F1307.7445&rft_id=info%3Adoi%2F10.4204%2FEPTCS.120.7&rft.aulast=Cuijpers&rft.aufirst=Pieter&rft_id=https%3A%2F%2Fdoi.org%2F10.4204%252FEPTCS.120.7&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrefix+order" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerlezCleavelandMarcus2014" class="citation book cs1">Ferlez, James; Cleaveland, Rance; Marcus, Steve (2014). "Generalized Synchronization Trees". <i>Foundations of Software Science and Computation Structures</i>. Lecture Notes in Computer Science. Vol. 8412. pp. 304–319. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-54830-7_20">10.1007/978-3-642-54830-7_20</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-54829-1" title="Special:BookSources/978-3-642-54829-1"><bdi>978-3-642-54829-1</bdi></a>.</cite><span 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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 class="mw-selflink selflink">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><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=Prefix_order&oldid=1193936345">https://en.wikipedia.org/w/index.php?title=Prefix_order&oldid=1193936345</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" 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