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Let $A$ be a finite set, $k\in\mathbb N$, $\tilde A\subset A^k$ and $\phi: \tilde A\to A$ be a map.

Consider the following property a set partition $P$ of $A$ might have: $$ \forall B_1,\dotsc,B_k\in P\; \exists B\in P\; \forall (b_1,\dotsc,b_k)\in (B_1\times\dotsb\times B_k)\cap\tilde A :\phi(b_1,\dotsc,b_k)\in B. $$ Put differently: the image of $\phi(B_1,\dotsc,B_k)$ is contained in a single block.

I would like to know whether this notion has (in some sense) a name, and whether there is a nice way to compute, given a set partition $Q$ of $A$, the finest set partition coarser than $Q$ having the above mentioned property.

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  • $\begingroup$ @SamHopkins quite right, thank you! The question comes from a practical computer problem, which I tried to abstract, and I made a mistake, obviously. I'll edit the condition. $\endgroup$
    – Martin Rubey
    Commented May 12, 2022 at 22:20
  • $\begingroup$ @bof, yes, thank you! $\endgroup$
    – Martin Rubey
    Commented May 13, 2022 at 7:25
  • $\begingroup$ I think the equivalence relation corresponding to the partition $P$ would be considered a congruence relation on the structure $(A,\phi)$ in the sense of universal algebra. $\endgroup$
    – bof
    Commented May 13, 2022 at 21:52

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