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# [[Transitive relation|Transitivity]]: if <math>a < b</math> and <math>b < c</math> then <math>a < c</math>.
Irreflexivity and transitivity together imply asymmetry, as does irreflexivity together with antisymmetry. Also, asymmetry implies irreflexivity. In other words, 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]].
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