Definition:Dual Ordering/Dual Ordered Set

From ProofWiki
Jump to navigation Jump to search


Let $\struct {S, \preceq}$ be an ordered set.

Let $\succeq$ be the dual ordering of $\preceq$.

The ordered set $\struct {S, \succeq}$ is called the dual ordered set (or just dual) of $\struct {S, \preceq}$.

That it indeed is an ordered set is a consequence of Dual Ordering is Ordering.

Also known as

A quite popular alternative for dual ordered set is opposite poset.

However, since this use conflicts with $\mathsf{Pr} \infty \mathsf{fWiki}$'s definition of a partially ordered set, dual ordered set is the name to be used.

Inverse ordered set can also be encountered.

Also see

  • Results about dual orderings can be found here.