# Subsemigroup of free semigroup that is not free

Is there any example of subsemigroups of a free semigroup, for exemple of $$(\mathbb{N}^*,\times)$$ generated by primes, that isn't free ?

I know that for free groups there is the Nielsen-Schreier theorem but on semigroup I read that it doesn't hold any more.

Consider $$(\mathbb{N}^*,+)$$ which is the free semigroup generated by one element, the subsemigroup generated by $$\{2,3\}$$ is not free:
It's not generated by one element (if it were, then $$1$$ would be that generator, but $$1$$ isn't in it), and for any two elements $$a,b$$ we have $$\underbrace{a+\cdots+a}_{b\text{ times}}=\underbrace{b+\cdots+b}_{a\text{ times}}$$, so it can't be the free semigroup generated by more than one element.