Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $f \in \hom_\mathbf{C}({\sim_X}, {\sim_Y})$?

share|improve this question

1 Answer 1

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

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.