Definition:Dual Isomorphism (Order Theory)

Definition
Let $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ be ordered sets.

Let $\phi:S \to T$ be a bijection.

Then $\phi$ is a dual isomorphism between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$ iff $\phi$ and $\phi^{-1}$ are decreasing mappings.

If there is a dual isomorphism between $\left({S, \preceq_S}\right)$ and $\left({T, \preceq_T}\right)$, then $\left({S, \preceq_S}\right)$ is dual to $\left({T, \preceq_T}\right)$.

Equivalently, $\left({S, \preceq_S}\right)$ is dual to $\left({T, \preceq_T}\right)$ iff $S$ with the dual ordering is isomorphic to $T$.