\begin{definition}[Definition:Dual Isomorphism (Order Theory)]
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 $\struct {T, \preceq_T}$ {{iff}} $\phi$ 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 $\struct {T, \preceq_T}$ {{iff}} $S$ with the dual ordering is isomorphic to $T$.
\end{definition}