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Let $X$ and $Y$ be two sets and $f: X \to Y$. Let $\sim$ be an equivalence relation on $X$. Please note that $f$ is not assumed to be compatible with $\sim$. Let $p: X \to X/\sim$ be the canonical projection to the quotient set. Let $r$ be a system of representatives, i.e. $r : X/\sim \to X$ such that, for any equivalence class $c \in X/\sim, r(c) \in c$.

Let $f_r = f \circ r \circ p$. Does someone know a standard terminology for this kind of object? I'd like to say something like "quotient of $f$ by $r$", "reduction of $f$ by $r$", etc.

Although I use this in a non-purely mathematical paper (about voting theory), I find it would be better to get inspiration from standard mathematical terminology if possible. Also, if there is no standard word for this, I would appreciate any suggestion that would help me not to break mathematical customs (for example, I suspect that "quotient" could be misleading).

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  • $\begingroup$ Restriction of $f$ to $r$? To give $r$ is to give its image. In fact if you work with the image of $f$ then it's the restriction of $f$ to this image. $\endgroup$
    – Kevin Buzzard
    Commented Jan 6, 2017 at 10:52
  • $\begingroup$ Thanks for your suggestion, which makes me realize that I oversimplified a bit my original question. In fact, what I consider exactly is $f_r : X \to Y$ defined by $f_r = f \circ r \circ []$, where the bracket operator takes the equivalence class of an element. For this reason, it seems to me that "restriction" does not convey exactly the notion I'm using. $\endgroup$
    – François Durand
    Commented Jan 6, 2017 at 11:20
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    $\begingroup$ Oh! Then I am going to go with "I don't know any particular standard terminology for this kind of object" (and I'd suggest that you edit the question, rather than fixing it in the comments). Maybe someone else can suggest something. $\endgroup$
    – Kevin Buzzard
    Commented Jan 6, 2017 at 11:32
  • $\begingroup$ Ok, done. Thanks again for your answers. $\endgroup$
    – François Durand
    Commented Jan 6, 2017 at 12:52

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