I know there is a splitting lemma for groups, but is there a similar lemma for semigroups or monoids? And do you have the proof of that?

You might look at Tilson's approach http://www.sciencedirect.com/science/article/pii/0022404987901083 where the kernel of a monoid morphism is a category and wreath product decompositions are obtained.

isno right notion of exactness for monoids. So maybe the answer is "mu" :). $\endgroup$ – Noah Schweber Jun 3 '14 at 23:05