Idempotent element of a semigroup ($e^2=e$)

An element of a semigroup is idempotent when multiplying it by itself leaves it unchanged. Every element of a left-zero or right-zero semigroup is idempotent, whereas a group has only its identity as an idempotent. The Ellis–Numakura lemma guarantees such an element in a nonempty compact one-sided topological semigroup. Additive idempotence on ultrafilters gives the idempotent ultrafilter relation $p+p=p$.