Definition:Dual Isomorphism (Order Theory)
Jump to navigation
Jump to search
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 $T$.
Sources
This article, or a section of it, needs explaining. In particular: section? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |