Definition:Dual Ordering/Dual Ordered Set
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Definition
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.
Sources
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.2$