The question is fairly dry: Is there any semigroup analogue of Lagrange's theorem for groups (counting as a generalization of the latter)? Let me guess the answer: Obviously yes. So the real question is: Any reference? Thank you in advance.

P.S.: I've notice of a Lagrange's theorem for Smarachande semigroups, but I would like to hear of different extensions, if possible (I don't think this is quite standard, but somebody defines a Smarachande semigroup to be any semigroup $(A, \star)$ for which there exist a proper subset $G$ of $A$, a unary operation $u: G \to G$ and a distinguished element $e \in G$ such that $(G, \star, u, e)$ is a group).