Characterization of split mono- and split epimorphisms of algebras

Given a concrete category $(\mathcal{A}, U : \mathcal{A} \to \mathsf{Set})$ of universal algebras, obviously if a morphism $f$ in $\mathcal{A}$ is split mono / split epi, then so is $U(f)$.

What is known about the converse in these concrete categories? Are there any known general statements characterizing split monos and split epis in a concrete category of universal algebras in terms of their underlying functions?

In $\mathsf{Grp}$ a morphism $f$ is split mono if and only, if $U(f)$ is injective and $\operatorname{Im} f$ has a normal complement in $B$.

In $\mathsf{Grp}$ a morphism $f$ is split epi if and only, if $U(f)$ is surjective and $\operatorname{Ker} f$ is a normal complement of some subgroup of $A$.

(these statements can be found here)

I can only speculate about possible generalizations.

• I doubt there's much to say here, insofar as it's already not totally trivial for groups, which are pretty simple as far as algebraic categories go. In particular the notion of complement is no good in monoids. – Kevin Carlson May 30 '16 at 20:25
• @KevinCarlson Any partial results are fine too. In particular, I guess I first need to collect more examples (i.e. characterizations for other varieties) to see whether there are any obvious patterns at all. – Stefan Perko May 30 '16 at 21:43