Definition:Dual Isomorphism (Order Theory)

Definition
Let $\struct {S, \preceq_S}$ and $\struct {T, \preceq_T}$ be ordered sets.

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

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

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

Equivalently, $\struct {S, \preceq_S}$ is dual to $\struct {T, \preceq_T}$ $S$ with the dual ordering is isomorphic to $T$.