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Given two relations $R\subseteq A\times B$ and $R’\subseteq A’\times B’$. Is it known/used that every relation $r\subseteq R\times R’$ can be characterized by two relations $\alpha\subseteq A\times A’$ and $\beta\subseteq B\times B’$ so that

$((a,b),(a’,b’))\in r \iff \Big((a,a’)\in\alpha\wedge (b,b’)\in\beta\wedge (a,b)\in R\implies (a’,b’)\in R’\Big)$

and if $R”\subseteq A”\times B”$, $\,r’\subseteq R’\times R”$, where

$r’$ is characterized by

$\alpha’\subseteq A’\times A”$ and $\beta’\subseteq B’\times B”$, then the composition $r’\circ r$ is characterized by the relations

$\alpha’\circ\alpha\subseteq A\times A”$ and

$\beta’\circ\beta\subseteq B\times B”$?

(Where $\circ$ denote the composition of relations). $\;$In diagram form:

$\require{AMScd}$

\begin{CD}

A@>\alpha>>A’@>\alpha’>>A”@. A@>\alpha’\circ\alpha>>A”\\

@VRVV r @VVR’ V r’@VVR”V \quad\implies\quad @VRVV r’\circ r @VVR”V\\

B@>>\beta>B’@>>\beta’>B” @. B@>>\beta’\circ\beta>B”

\end{CD}

Does this category of relations as objects and relation between relations as morphisms have a name?

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The reason for my interest in this category:

Suppose $A=B\times B$ and that $R\subseteq A\times B$ is the composition in a magma. Then the functions among the morphisms between two such objects defines magma morphisms $B\to B’$.

Suppose $B=\mathcal P(A)$ and that $R\subseteq A\times B$ is the relation

$(a,S)\in R\iff a\in\overline{S}$ for some topology on $A$. Then the functions among the morphisms between two such objects define continuous functions $A\to A’$.

Edit:

The category of relations as objects and pairs of relations defined by some $\alpha,\beta$ (as above) as morphisms, is a subcategory of the category of relations as objects and relations of relations as morphisms. Very few relations between relations can be defined with two relations $\alpha,\beta$, but those relations which can seems to be an interesting category.

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Such a characterisation of ‘relations between relations’ is not possible.

As a counterexample, let $A,B,A’,B’,\alpha,\beta$ be whatever you want them to be, let $R = \varnothing$, let $R’ = A’ \times B’$. Then:

- $(a’,b’) \in R’$ for all $a’ \in A’$ and $b’ \in B’$, so the statement on the right-hand side of your $\Leftrightarrow$ symbol is true for all $a,b,a’,b’$.
- $R \times R’ = \varnothing$, so the only relation $r \subseteq R \times R’$ is the empty relation. Hence the left-hand side of your $\Leftrightarrow$ symbol is false $a,b,a’,b’$.

I can’t answer your other questions, since they are contingent on such a characterisation of ‘relations between relations’ being possible.

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