Isomorphism between finitely generated semigroups

Question

May I ask you to make light on the following point. I know that there maybe certain conditions of the type of two semigroups. I am simply looking articles or a nice hint-answer just for sure:

Does any isomorphism map generators to generators when we consider finitely generated semigroups?

I wonder what would it be if they are finitely presented?

Thanks for any hint.

Answer 1

Let $f:S \to T$ be a semigroup isomorphism. Then $E$ is a generating set for $S$ if, and only if, $f(E)$ is a generating set for $T$. No need for $S$ to be finitely generated or finitely presented.