In the paper of Alan J.Cain et al " Green index in semigroups: Generators, presentations, and automatic structures", is mentioned that if $T$is a finitely generated subsemigroup of a finitely generated(f.g)semigroup $S$, then if the growth function of $S$ is polynomially upper-bounded so will be the growth function of $T$. Also there is mentioned that in general the converse is not true! Can someone please explain this to me by an example, why this is not true?
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