Then, as for your case, the composition of two partial functions (from $X$ to $X$) makes always sense. Thanks for the clarification

Thank you, Benjamin. But I'm not quite sure to understand what you mean by saying that the composition of partial functions (from $X$ to itself) could make no sense in some cases. Partial functions (between sets) should be composed with each other in the same way as binary relations. Then, the corresponding category would be equivalent to the usual category of pointed sets.

As for the rest, 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 the -cypa- ones where the theorem sounds true by definition. I know, non-trivial and interesting are not well-defined terms. But let me believe in your common sense.

Yes, I mean that theorem. But Lagrange's theorem for Smarandache semigroups is not exactly what you're thinking of. In particular, it is not a statement about all the subsemigroups of a given semigroup. Indeed, it is not even about arbitrary Smarandache semigroups. Instead, it concerns a special class of them, which somebody calls Smarandache lagrangian semigroups. Thus, what they refer to as the Lagrange's theorem for Smarachande semigroups is essentially true by definition. And yes, I agree with you that this may be a -cypa- thing, but I'm not here to quibble in the value of others' work.

Sorry, I was a little bit sloppy. I'd better say that the kind of algebras considered by Yemon are reminiscent of quasi-normed algebras. Let me fix it.