partial order

A partial order is a homogeneous relation that is reflexive, transitive, and antisymmetric.

Formally,
a binary relation $\le$ over a set $S$ is a partial order if for all $a,b,c$ in $S$,

• $a\le a$ (reflexive)
• if $a\le b$ and $b\le c$ then $a\le c$ (transitive)
• if $a\le b$ then $b\nleqq a$ (antisymmetric)

A partial order is an antisymmetric preorder.

A partial order that is connected, i.e. where every pair is comparable, is a total order.