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Given a function $f : X \to Y$, an equivalence relation $\sim_X$ on $X$ and an equivalence relation $\sim_Y$ on $Y$, there is a notion of ``compatibility'' between $f$, $\sim_X$ and $\sim_Y$ if the following holds: $x_0 \mathbin{\sim_X} x_1 \implies f(x_0) \mathbin{\sim_Y} f(x_1)$. If it holds, we can define a quotient over $f$ as the function $f/(\sim_X, \sim_Y) : X/{\sim_X} \to Y/{\sim_Y}$ that maps each $E \in X/{\sim_X}$ to the unique element of $Y/{\sim_Y}$ that contains $f(E)$.

Is this notion (of compatibility and quotient of functions) already well-studied, and is there a proper term for this ``compatibility''? Perhaps $f$ is some kind of homomorphism...

In the special case where ${\sim_X} = {\sim_Y}$ and $X = Y$, we can say that $\sim_X$ is a congruence relation that is invariant under $f$.

Update: Wikipedia calls such an $f$ a morphism from $\sim_X$ to $\sim_Y$. However, as I am using this concept together with other categories and morphisms, such terminology can be confusing. Is there a standard name for the category $\mathbf{C}$ of equivalence relations (or partitions, I suppose) and such morphisms between them, so I can state without ambiguity that $f \in \hom_\mathbf{C}({\sim_X}, {\sim_Y})$?

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See, for example, page 118 of Bourbaki's "Theory of Sets Volume 1."

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Thanks for the reference, but doesn't it only account for the $\sim_X$ part? –  Herng Yi Nov 14 '12 at 8:51

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