What structure has been found for functions with this relationship. 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.

 A: This question reminds me of Peter Freyd's work on fixed points.  I forget the reference for that but Google will show you some of the follow-up work that other people have done since.
Fixed points of this nature live in domains, which live in categories that do not have pullbacks and equalisers, so the notion of kernel is not going to help you.
Be warned, however: there are results in this area that are extremely clever tricks, but such tricks are possibly only discovered once per decade.  In the meantime, you will cover thousands of pages with calculations that lead to absolutely nothing.
The two examples I had in mind were Pataraia's and Beki&cacute;'s fixed point theorems. (The late Georgian Dito) Pataraia's theorem is that in a DCPO $P$ with $\bot$, every monotone $f:P\to P$ has a least fixed point. As the comments indicate, the proof is very clever, very short and available in various places on the Web, though Pataraia never published it in a journal.
A: Let me advertise clone theory.
Pick an underlying set A,and consider a family of functions f_i from (powers of) A to A, perhaps
of different arities.  If this family contains the projections (like the primitive recursive functions do)
and is closed under function composition (as is the case with PR functions), the family is a clone of
A.
One can study clones which preserve finitary relations R (so R becomes a subalgebra of A^n when
augmented by this family of functions), as well as commuting functions, as well as studying relations
between their kernels.
I think (from my limited point of view) a proper study of recursive clone theory has yet to be
undertaken, and that your question fits into such a study.
The only published source I have read on clone theory exists in chapters 3 and 4 of "Algebras, Lattices,
Varieties" by McKenzie, McNulty, and Taylor.  There may be more appropriate treatments in the computer
science literature, but you might find the general algebraic perspective of value.
Gerhard "Ask Me About System Design" Paseman, 2013.06.13
