Editing Partially ordered set (section)

== Alternative definitions ==

Another way of defining a partial order, found in [[computer science]], is via a notion of [[Comparability|comparison]]. Specifically, given <math>\leq, <, \geq, \text{ and } ></math> as defined previously, it can be observed that two elements ''x'' and ''y'' may stand in any of four [[mutually exclusive]] relationships to each other: either {{nowrap|''x'' < ''y''}}, or {{nowrap|1=''x'' = ''y''}}, or {{nowrap|''x'' > ''y''}}, or ''x'' and ''y'' are ''incomparable''. This can be represented by a function <math>\text{compare}: P \times P \to \{<,>,=,\vert \}</math> that returns one of four codes when given two elements.<ref>{{cite web |title=Finite posets |url=http://match.stanford.edu/reference/combinat/sage/combinat/posets/posets.html#sage.combinat.posets.posets.FinitePoset.compare_elements |website=Sage 9.2.beta2 Reference Manual: Combinatorics |access-date=5 January 2022|quote=compare_elements(''x'', ''y''): Compare ''x'' and ''y'' in the poset. If {{nowrap|''x'' < ''y''}}, return −1. If {{nowrap|1=''x'' = ''y''}}, return 0. If {{nowrap|''x'' > ''y''}}, return 1. If ''x'' and ''y'' are not comparable, return None.}}</ref><ref>{{cite tech report |last1=Chen |first1=Peter |last2=Ding |first2=Guoli |last3=Seiden |first3=Steve |title=On Poset Merging |page=2 |url=https://www.math.lsu.edu/~ding/poset.pdf |access-date=5 January 2022 |quote=A comparison between two elements s, t in S returns one of three distinct values, namely s≤t, s>t or s<nowiki>|</nowiki>t.}}</ref> This definition is equivalent to a ''partial order on a [[setoid]]'', where equality is taken to be a defined [[equivalence relation]] rather than set equality.<ref>{{cite conference |conference=CALCULEMUS-2003 – 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning|location=Roma, Italy |date=11 September 2003 |url=https://hal.science/hal-02549766/document#page=98 |publisher=Aracne |language=en|title=Making proofs in a hierarchy of mathematical structures|first1=Virgile|last1=Prevosto|first2=Mathieu|last2=Jaume|pages=89–100}}</ref>

Wallis defines a more general notion of a ''partial order relation'' as any [[homogeneous relation]] that is [[Transitive relation|transitive]] and [[Antisymmetric relation|antisymmetric]]. This includes both reflexive and irreflexive partial orders as subtypes.<ref name=Wallis>{{cite book |last1=Wallis |first1=W. D. |title=A Beginner's Guide to Discrete Mathematics |date=14 March 2013 |publisher=Springer Science & Business Media |isbn=978-1-4757-3826-1 |page=100 |url=https://books.google.com/books?id=ONgRBwAAQBAJ&dq=%22partial%20order%20relation%22&pg=PA100 |language=en}}</ref>

A finite poset can be visualized through its [[Hasse diagram]].<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |author-link2=Howard Ensign Simmons Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=https://archive.org/details/topologicalmetho00merr/page/28 |access-date=27 July 2012 |pages=[https://archive.org/details/topologicalmetho00merr/page/28 28] |quote=A partially ordered set is conveniently represented by a ''Hasse diagram''... |url-access=registration }}</ref> Specifically, taking a strict partial order relation <math>(P,<)</math>, a [[directed acyclic graph]] (DAG) may be constructed by taking each element of <math>P</math> to be a node and each element of <math> < </math> to be an edge. The [[transitive reduction]] of this DAG{{efn|which always exists and is unique, since <math>P</math> is assumed to be finite}} is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.