Definition:Dual Isomorphism (Order Theory)

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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 $\struct {T, \preceq_T}$ if and only if $\phi$ 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 $\struct {T, \preceq_T}$ if and only if $S$ with the dual ordering is isomorphic to $T$.


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