# Splitting lemma for semigroups or monoids

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?

• Well, I'm not sure what exactness ought to mean in the context of monoids, since the notion of "kernel" is a bit odd - for example, consider the monoid $N^+$ of natural numbers (including 0) together with $\infty$, and the monoid $T=(0, \infty)$. Then there is a homomorphism from $N^+$ to $T$ taking 0 to 0 and everything else to $\infty$ - clearly this map is not injective, but the "kernel" (preimage of 0) is $\{0\}$. I don't know, but I suspect that the "right" notion of exactness for semigroups/monoids is one which directly implies a splitting lemma. – Noah Schweber Jun 3 '14 at 23:05
• (cont'd) See, however, mathoverflow.net/questions/83080/exact-sequence-of-monoids, where JBorger argues that there is no right notion of exactness for monoids. So maybe the answer is "mu" :). – Noah Schweber Jun 3 '14 at 23:05