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== Notation ==
Given a set <math>P</math> and a partial order relation, typically the non-strict partial order <math>\leq</math>, we may uniquely extend our notation to define four partial order relations <math>\leq,</math> <math><,</math> <math>\geq,</math> and <math>></math>, where <math>\leq</math> is a non-strict partial order relation on <math>P</math>, <math> < </math> is the associated strict partial order relation on <math>P</math> (the [[irreflexive kernel]] of <math>\leq</math>), <math>\geq</math> is the dual of <math>\leq</math>, and <math> > </math> is the dual of <math> < </math>. Strictly speaking, the term ''partially ordered set'' refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, <math>(P,\leq)</math> or <math>(P,<)</math>, or, in rare instances, the non-strict and strict relations together, <math>(P,\leq,<)</math>.<ref>{{cite book |last1=Avigad |first1=Jeremy |last2=Lewis |first2=Robert Y. |last3=van Doorn |first3=Floris |title=Logic and Proof |date=29 March 2021 |edition=Release 3.18.4 |url=https://leanprover.github.io/logic_and_proof/relations.html#more-on-orderings |access-date=24 July 2021 |chapter=13.2. More on Orderings |quote=So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one. |archive-date=3 April 2023 |archive-url=https://web.archive.org/web/20230403074506/https://leanprover.github.io/logic_and_proof/relations.html#more-on-orderings |url-status=dead }}</ref>
The term ''ordered set'' is sometimes used as a shorthand for ''partially ordered set'', as long as it is clear from the context that no other kind of order is meant. In particular, [[Total order|totally ordered sets]] can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than <math>\leq</math> such as <math>\sqsubseteq</math><ref>{{cite web |last1=Rounds |first1=William C. |title=Lectures slides |url=http://www.eecs.umich.edu/courses/eecs203-1/203-Mar7.pdf |website=EECS 203: DISCRETE MATHEMATICS |access-date=23 July 2021 |date=7 March 2002}}</ref> or <math>\preceq</math><ref>{{cite book |last1=Kwong |first1=Harris |title=A Spiral Workbook for Discrete Mathematics |date=25 April 2018 |url=https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.04%3A_Partial_and_Total_Ordering |access-date=23 July 2021 |language=en |chapter=7.4: Partial and Total Ordering}}</ref> to distinguish partial orders from total orders.
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If the count is made only [[up to]] isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... {{OEIS|A000112}} is obtained.
==
A poset <math>P^*=(X^*, \leq^*)</math> is called a '''subposet''' of another poset <math>P=(X, \leq)</math> provided that <math>X^*</math> is a [[subset]] of <math>X</math> and <math>\leq^*</math> is a subset of <math>\leq</math>. The latter condition is equivalent to the requirement that for any <math>x</math> and <math>y</math> in <math>X^*</math> (and thus also in <math>X</math>), if <math>x \leq^* y</math> then <math>x \leq y</math>.
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In [[computer science]], algorithms for finding linear extensions of partial orders (represented as the [[reachability]] orders of [[directed acyclic graph]]s) are called [[topological sorting]].
== In
{{main|Posetal category}}
Every poset (and every [[Preorder|preordered set]]) may be considered as a [[Category (mathematics)|category]] where, for objects <math>x</math> and <math>y,</math> there is at most one [[morphism]] from <math>x</math> to <math>y.</math> More explicitly, let {{nowrap|1=hom(''x'', ''y'') = {{mset|(''x'', ''y'')}}}} if {{nowrap|''x'' ≤ ''y''}} (and otherwise the [[empty set]]) and <math>(y, z) \circ (x, y) = (x, z).</math> Such categories are sometimes called ''[[Posetal category|posetal]]''.
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Posets are [[Equivalence of categories|equivalent]] to one another if and only if they are [[Isomorphism of categories|isomorphic]]. In a poset, the smallest element, if it exists, is an [[initial object]], and the largest element, if it exists, is a [[terminal object]]. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is [[isomorphism-closed]].
== Partial orders in
{{Main|Partially ordered space}}
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