I know the standard definition, but are there any alternative definitions? In particular, I distinctly remember seeing a remark that the definition given by John Kelley in his classic text on General Topology was non-standard.
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Just to clarify, which do you refer to as the "standard definition", and do you still have Kelley's text on hand? – Brevan Ellefsen Nov 05 '18 at 21:45
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Kelley, p. 13: "An ordering (partial ordering, quasi-ordering) is a transitive relation". – GEdgar Nov 05 '18 at 21:55
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@GEdgar: Thanks. That, in reality, is EXACTLY what I was looking for. – Nov 05 '18 at 23:07
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So a poset is a reflexive, antisymmetric and transitive relation:
$x\leq x$ => Reflexive
$x\leq y\land y\leq x\;\Rightarrow\;x=y$ => Antisymmetric
$x\leq y\land y\leq z\;\Rightarrow\;x\leq z$ => Transitive
...for $x,y,z\in M$, whereas the inverse relation still a poset is:
${\displaystyle y\preceq x:\Longleftrightarrow x\leq y}$
This is the fundamental idea and should be the original version (with different variables of course)
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