Re: M times M = M?

Yes, because if L is the lanaguage of M then the question becomes "L
intersect L = L?". But is there other ways to prove this?

I believe what you're looking for is the notion of isomorphism.

You can try to map the states and transitions of M to those of MxM
in such a way that the transitions between the mapped states are
exactly the mappings of the transitions between their originals.

--
Reinier
.