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. 1.

   Reflexivity: $a\leq a$ for all $a\in A$

2. 2.

   Antisymmetry: If $a\leq b$ and $b\leq a$ for any $a,b\in A$, then $a=b$

3. 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 $\leq$ is defined as a transitive irreflexive binary relation (on a set). As a result, if $a\leq b$, then $b\nleq a$, and therefore $\leq$ is antisymmetric.