= Idempotent element of a semigroup
{title2=$e^2=e$}
= Semigroup idempotent
{synonym}
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 $p+p=p$.
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