I've thought about finest equivalence relations and came up with a conjecture but I am neither able to prove nor able to disprove it. A hint would be great.
Be $M$ a set, be $f$ a bijective function with $f: M \rightarrow M$ and be $\sim$ the finest equivalence relation on the set $M$ with $x \sim f(x)$ for all $x \in M$.
Be also $Z$ a set and $g$ a surjective function with $g: M \rightarrow Z$ and $\forall x \in M: g(x) = g(f(x))$
Does this imply: $$\forall x,y \in M: x \sim y \implies g(x) = g(y)$$?
Define a relation $\approx$ on $M$ by $x \approx y$ if $g(x) = g(y)$.
Can you see that $\approx$ is an equivalence relation on $M$ and that $x \approx f(x)$ for all $x \in M$?