weak order

A weak order is a homogeneous relation that is reflexive, transitive, and connected.

Formally,
a binary relation $\le$ over a set $S$ is a weak 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)
• either $a\le b$ or $b\le a$ (connected)

A weak order is a strongly connected preorder.