partial order

A partial order (often simply referred to as an order or ordering) is a relation $\leq\:\subset A\times A$ that satisfies the following three properties:

1. Reflexivity: $a \leq a$ for all $a\in A$
2. Antisymmetry: If $a \leq b$ and $b \leq a$ for any $a, b\in A$ , then $a = b$
3. Transitivity: If $a \leq b$ and $b \leq c$ for any $a, b, c\in A$ , then $a \leq c$

A total order is a partial order that satisfies a fourth property known as comparability:

• Comparability: For any $a,b\in A$ , either $a\leq b$ or $b\leq a$ .

A set and a partial order on that set define a poset.

Remark. In some literature, especially those dealing with the foundations of mathematics, a partial order $\le$ is defined as a transitive irreflexive binary relation (on a set). As a result, if $a\le b$ , then $b \nleq a$ , and therefore $\le$ is antisymmetric.