Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, $S\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$, and $T\colon C\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}D$ be relations.
-
Interaction With Ranges and Domains. We have
\begin{align*} \mathrm{dom}\webleft (S\mathbin {\diamond }R\webright ) & \subset \mathrm{dom}\webleft (R\webright ),\\ \mathrm{range}\webleft (S\mathbin {\diamond }R\webright ) & \subset \mathrm{range}\webleft (S\webright ). \end{align*}
-
Associativity. We have
\[ \webleft (T\mathbin {\diamond }S\webright )\mathbin {\diamond }R = T\mathbin {\diamond }\webleft (S\mathbin {\diamond }R\webright ). \]
-
Unitality. We have
\begin{align*} \chi _{B}\mathbin {\diamond }R & = R,\\ R\mathbin {\diamond }\chi _{A} & = R. \end{align*}
-
Interaction With Inverses. We have
\[ \webleft (S\mathbin {\diamond }R\webright )^{\dagger } = R^{\dagger }\mathbin {\diamond }S^{\dagger }. \]
-
Interaction With Composition. We have
\begin{align*} \chi _{B} & \subset R\mathbin {\diamond }R^{\dagger },\\ \chi _{A} & \subset R^{\dagger }\mathbin {\diamond }R. \end{align*}
Item 1: Interaction With Ranges and Domains
Clear.
Indeed, we have
\begin{align*} \webleft (T\mathbin {\diamond }S\webright )\mathbin {\diamond }R & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\int ^{y\in C}T^{-_{1}}_{x}\times S^{x}_{-_{2}}\webright )\mathbin {\diamond }R\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{x\in B}\webleft (\int ^{y\in C}T^{-_{1}}_{x}\times S^{x}_{y}\webright )\mathbin {\diamond }R^{y}_{-_{2}}\\ & = \int ^{x\in B}\int ^{y\in C}\webleft (T^{-_{1}}_{x}\times S^{x}_{y}\webright )\mathbin {\diamond }R^{y}_{-_{2}}\\ & = \int ^{y\in C}\int ^{x\in B}\webleft (T^{-_{1}}_{x}\times S^{x}_{y}\webright )\mathbin {\diamond }R^{y}_{-_{2}}\\ & = \int ^{y\in C}\int ^{x\in B}T^{-_{1}}_{x}\times \webleft (S^{x}_{y}\mathbin {\diamond }R^{y}_{-_{2}}\webright )\\ & = \int ^{x\in B}T^{-_{1}}_{x}\times \webleft (\int ^{y\in C}S^{x}_{y}\mathbin {\diamond }R^{y}_{-_{2}}\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{x\in B}T^{-_{1}}_{x}\times \webleft (S\mathbin {\diamond }R\webright )^{x}_{-_{2}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}T\mathbin {\diamond }\webleft (S\mathbin {\diamond }R\webright ). \end{align*}
In the language of relations, given $a\in A$ and $d\in D$, the stated equality witnesses the equivalence of the following two statements:
-
We have $a\sim _{\webleft (T\mathbin {\diamond }S\webright )\mathbin {\diamond }R}d$, i.e. there exists some $b\in B$ such that:
-
We have $a\sim _{R}b$;
-
We have $b\sim _{T\mathbin {\diamond }S}d$, i.e. there exists some $c\in C$ such that:
-
We have $b\sim _{S}c$;
-
We have $c\sim _{T}d$;
-
We have $a\sim _{T\mathbin {\diamond }\webleft (S\mathbin {\diamond }R\webright )}d$, i.e. there exists some $c\in C$ such that:
-
We have $a\sim _{S\mathbin {\diamond }R}c$, i.e. there exists some $b\in B$ such that:
-
We have $a\sim _{R}b$;
-
We have $b\sim _{S}c$;
-
We have $c\sim _{T}d$;
both of which are equivalent to the statement
- There exist $b\in B$ and $c\in C$ such that $a\sim _{R}b\sim _{S}c\sim _{T}d$.
Indeed, we have
\begin{align*} \chi _{B}\mathbin {\diamond }R & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{x\in B}\webleft (\chi _{B}\webright )^{-_{1}}_{x}\times R^{x}_{-_{2}}\\ & = \bigvee _{x\in B}\webleft (\chi _{B}\webright )^{-_{1}}_{x}\times R^{x}_{-_{2}}\\ & = \bigvee _{\substack {x\in B\\ x=-_{1}
}}R^{x}_{-_{2}}\\ & = R^{-_{1}}_{-_{2}}, \end{align*}
and
\begin{align*} R\mathbin {\diamond }\chi _{A} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{x\in A}R^{-_{1}}_{x}\times \webleft (\chi _{A}\webright )^{x}_{-_{2}}\\ & = \bigvee _{x\in B}R^{-_{1}}_{x}\times \webleft (\chi _{A}\webright )^{x}_{-_{2}}\\ & = \bigvee _{\substack {x\in B\\ x=-_{2}
}}R^{-_{1}}_{x}\\ & = R^{-_{1}}_{-_{2}}. \end{align*}
In the language of relations, given $a\in A$ and $b\in B$:
- The equality
\[ \chi _{B}\mathbin {\diamond }R=R \]
witnesses the equivalence of the following two statements:
-
We have $a\sim _{b}B$.
-
There exists some $b'\in B$ such that:
-
We have $a\sim _{R}b'$
-
We have $b'\sim _{\chi _{B}}b$, i.e. $b'=b$.
- The equality
\[ R\mathbin {\diamond }\chi _{A}=R \]
witnesses the equivalence of the following two statements:
-
There exists some $a'\in A$ such that:
-
We have $a\sim _{\chi _{B}}a'$, i.e. $a=a'$.
-
We have $a'\sim _{R}b$
-
We have $a\sim _{b}B$.
Item 4: Interaction With Inverses
Clear.
Item 5: Interaction With Composition
Clear.
