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I’m dealing with relations on relations $\mathcal{R} \subseteq \mathcal{P}(A \times A) \times \mathcal{P}(A \times A)$ that have the following properties:

  • $(R_{1}, S_{1}) \in \mathcal{R} \mathrel\wedge (R_{2}, S_{2}) \in \mathcal{R} \rightarrow (R_{1} \circ R_{2}, S_{1} \circ S_{2}) \in \mathcal{R}$

  • $(\forall i \in I. (R_{i}, S_{i}) \in \mathcal{R}) \rightarrow \bigl(\bigcup_{i \in I} R_{i}, \bigcup_{i \in I} S_{i}\bigr) \in \mathcal{R}$

Is there a common name for such relations?

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    $\begingroup$ You can view the set of relations as a quantale and these would be quantale congruences. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 15, 2019 at 16:49
  • $\begingroup$ That’s interesting. Thanks for the hint. However, wouldn’t $\mathcal{R}$ need to be an equivalence relation in order to be a congruence? In my use case, $\mathcal{R}$ typically isn’t symmetric and might not even be reflexive or transitive. $\endgroup$
    – Wolfgang Jeltsch
    Commented Dec 15, 2019 at 17:07
  • $\begingroup$ Sorry. I missed it wasn't an equivalence relation. I'm sure quantale people have a name for that $\endgroup$
    – Benjamin Steinberg
    Commented Dec 15, 2019 at 17:11

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