Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ and $S\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$ be relations.
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Functoriality. The assignment $R\mapsto R^{\dagger }$ defines a functor (i.e. morphism of posets)
\[ \webleft (-\webright )^{\dagger }\colon \mathbf{Rel}\webleft (A,B\webright )\to \mathbf{Rel}\webleft (B,A\webright ). \]
In particular, given relations $R,S\colon A\mathrel {\rightrightarrows \kern -9.5pt\mathrlap {|}\kern 6pt}B$, we have:
- If $R\subset S$, then $R^{\dagger }\subset S^{\dagger }$.
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Interaction With Ranges and Domains. We have
\begin{align*} \mathrm{dom}\webleft (R^{\dagger }\webright ) & = \mathrm{range}\webleft (R\webright ),\\ \mathrm{range}\webleft (R^{\dagger }\webright ) & = \mathrm{dom}\webleft (R\webright ). \end{align*}
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Interaction With Composition I. We have
\[ \webleft (S\mathbin {\diamond }R\webright )^{\dagger } = R^{\dagger }\mathbin {\diamond }S^{\dagger }. \]
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Interaction With Composition II. We have
\begin{align*} \chi _{B} & \subset R\mathbin {\diamond }R^{\dagger },\\ \chi _{A} & \subset R^{\dagger }\mathbin {\diamond }R. \end{align*}
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Invertibility. We have
\[ \webleft (R^{\dagger }\webright )^{\dagger } = R. \]
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Identity. We have
\[ \chi ^{\dagger }_{A} = \chi _{A}. \]
