strict weak order

A strict weak order is a relation that is irreflexive, transitive, and antisymmetric, where the induced incomparability relation is also transitive.

Formally, a binary relation $<$ is a strict weak order over a set $X$ if for all $a,b,c$ in $X$,

• $a\nless a$ (irreflexive)
• If $a<b$ then $b\nless a$ (antisymmetric)
• If $a<b$ and $b<c$ then $a<c$ (transitive)
• If $a,b$ incomparable and $b,c$ incomparable then $a,c$ incomparable (transitive${}^{†}$)

A strict weak order where $a<b$ defines a weak order by the converse of the complement, so $b\le a$ whenever $a<b$ is false.