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 Smarandache semigroups, but I would like to hear of different extensions, if possible (I don't think this is quite standard, but somebody defines a Smarandache 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).

**Edit.** This is basically a comment to the subsequent answer of Vladimir Dotsenko. Let me highlight that I'm not asking for (possible) extensions to *arbitrary* semigroups. And I don't expect that, if any *non-trivial* extension is possible, it looks exactly like Lagrange's theorem for groups. 

> I'm just asking for any possible
> *non-trivial* extension that is already
> there, in the literature. Say, for
> instance, an extension to some
> *interesting* classes of semigroups (apart from groups and those where the theorem sounds true by definition, e.g. Smarandache lagrangian semigroups).

I know, *non-trivial* and *interesting* are not well-defined terms. But I have faith in your common sense.