# Partially ordered set

A partially ordered set, or poset, is a structure that formalizes and generalizes orderings on a set.

## Definition

A partially ordered set is a set, S , with a binary relation, ${\displaystyle \leq }$, that satisfies the following properties:

Reflexivity
${\displaystyle x\leq x}$ for all ${\displaystyle x\in S}$.
Transitivity
If ${\displaystyle x\leq y}$ and ${\displaystyle y\leq z}$ then ${\displaystyle x\leq z}$, for all ${\displaystyle x,y,z\in S}$.
Antisymmetry
If ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x}$ then ${\displaystyle x=y}$, for all ${\displaystyle x,y\in S}$.

Two elements of a poset, x  and y , are said to be comparable if ${\displaystyle x,y\in S}$ then either ${\displaystyle x\leq y}$ or ${\displaystyle y\leq x}$.

## Totally ordered set

A totally ordered set is a partially ordered set such that all elements are comparable with all other elements. That is if ${\displaystyle x,y\in S}$ then either ${\displaystyle x\leq y}$ or ${\displaystyle y\leq x}$.