Partial Order

algebra set-theory

Definition

Partial Order

A partial order is a binary relation $\le$ on a set $S$ that is reflexive, antisymmetric, and transitivity. A set equipped with a partial order is termed a partially ordered set or poset.

Axioms

Reflexivity: For all $a\in S$, it holds that $a\le a$.

Antisymmetry: For all $a,b\in S$, if $a\le b$ and $b\le a$, then $a=b$.

Transitivity: For all $a,b,c\in S$, if $a\le b$ and $b\le c$, then $a\le c$.