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 is finite and , then certainly is finite. So every element of has finite order, and thus is finite.
However, now consider the semigroup . Certainly is not finite, but now every element is idempotent, so that . So is periodic.