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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.

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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.

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