# Definition:Dual Ordering/Dual Ordered Set

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## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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

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

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 ProofWiki's definition of a partially ordered set, dual ordered set is the name to be used.

Inverse ordered set can also be encountered.