## 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.