Editing Partially ordered set (section)

=== Strict partial orders ===

An '''irreflexive''', '''strong''',<ref name=Wallis/> or '''{{visible anchor|strict partial order|Strict partial order|Irreflexive partial order}}''' is a homogeneous relation < on a set <math>P</math> that is [[Irreflexive relation|irreflexive]], [[Asymmetric relation|asymmetric]] and [[Transitive relation|transitive]]; that is, it satisfies the following conditions for all <math>a, b, c \in P:</math>
# [[Irreflexive relation|Irreflexivity]]:  <math>\neg\left( a < a \right)</math>, i.e. no element is related to itself (also called anti-reflexive).
# [[Asymmetric relation|Asymmetry]]: if <math>a < b</math> then not <math>b < a</math>.
# [[Transitive relation|Transitivity]]: if <math>a < b</math> and <math>b < c</math> then <math>a < c</math>.

A transitive relation is asymmetric if and only if it is irreflexive.<ref name="Flaška 2007">{{cite journal |last1=Flaška |first1=V. |last2=Ježek |first2=J. |last3=Kepka |first3=T. |last4=Kortelainen |first4=J. |title=Transitive Closures of Binary Relations I |journal=Acta Universitatis Carolinae. Mathematica et Physica |year=2007 |volume=48 |issue=1 |pages=55–69 |publisher=School of Mathematics – Physics Charles University |location=Prague |url=http://dml.cz/dmlcz/142762 }} Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".</ref> So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

A strict partial order is also known as an asymmetric [[strict preorder]].