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→Strict partial orders: Trying this out again. I think it should be made clear that there are two required conditions, unlike, e.g., the situation for a non-strict partial order. Tag: Reverted |
Undid revision 1255348218 by Holderbp (talk) This fact was clear in the previous version; it is not in fact more important than clearly articulating what the definition actually is |
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=== 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 [[Transitive relation|transitive]], [[Irreflexive relation|irreflexive]], and [[Asymmetric relation|asymmetric]]
# [[Transitive relation|Transitivity]]: if <math>a < b</math> and <math>b < c</math> then <math>a < c</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>.
A transitive relation is asymmetric if and only if it is irreflexive
A strict partial order is also known as an asymmetric [[strict preorder]].
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