Partial order

A partially-ordered set (or poset, for short) is one of the more basic kinds of orderings, whose requirements are more strict than those of a preorder, but looser than those of a total order.

Definition

A partial ordering on a set $S$ is a relation $⩽$ satisfying the following properties:

reflexivity

for any $x\in S$, $x⩽x$;

transitivity

for any $x,y,z\in S$, $x⩽y$ and $y⩽z$ together imply $x⩽z$;

antisymmetry

if $x⩽y$ and $y⩽x$, then $x=y$.