What is a poset?

Question

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.

Answer 1

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)