Finitely generated Clifford semigroup

Question

If $S$ is a finitely generated Clifford semigroup, therefore its corresponding semilattice is finitely generated. We know that a finitely generated semilattice is finite. Does this mean that $S$ is finite in this case? Can someone please help me to understand this?

Answer 1

You can't because it isn't true that a subsemigroup of a Clifford semigroup is finitely generated. For example, every group is a Clifford semigroup, and so, say, the free group with two generators is finitely generated but has non-finitely generated subgroups, and hence subsemigroups.