Every finite semigroup is periodic, but not every periodic semigroup is finite

Show that every finite semigroup is periodic, but not every periodic semigroup is finite.


Recall that a semigroup is called periodic if every element has finite order, meaning that the cyclic subsemigroup it generates is finite.

If S is finite and s \in S, then certainly \langle s \rangle \subseteq S is finite. So every element of S has finite order, and thus S is finite.

However, now consider the semigroup S = (\mathbb{N},\mathsf{max}). Certainly S is not finite, but now every element s \in S is idempotent, so that \langle s \rangle = \{s\}. So S is periodic.

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