Given $f$ and $g$

$\forall x y. f(x) = f(y) \Longrightarrow f(g(x)) = f(g(y))$

Or equivalently

$ker\ f \subseteq ker\ (f \circ g)$.

Note: if $f$ is injective then this holds for any $g$.

Explanation/motivation: I'm a CompSci who is trying to become more theoretical, and have been playing with recursive functions. I've discovered that a useful property about fixpoints is entailed by the above property, (namely that $\exists h. f(\mu g) = \mu h$,) and I'm wondering:

- What has been discovered around functions of this shape.
- More generally, what area of maths should I be investigating to learn more.