Idempotents in compact semigroups

Question

Hi, I've read somewhere that a compact (Abelian) topological semigroup has at least one idempotent and if it contains identity and is not a group then it has one extra idempotent apart from the identity. Could you please give me some reference, where I could find this? Thanks in advance for any help.

Answer 1

Let me supplement Boris's answer since the result you ask for is easier than the result Boris cites.

Let $T$ be a (nonempty) semigroup. Then an easy exercise is the following: $T$ is a group iff $tT=T=Tt$ for all $t\in T$.

Suppose now that $S$ is a compact semigroup. Then by Zorn's lemma and compactness it contains a minimal closed subsemigroup $T$. By minimality and continuity of multiplication $tT=T=Tt$ for all $t\in T$ and hence $T$ is a group. Thus $T$ has an idempotents (its identity) and so $S$ has an idempotent.

Suppose now that $S$ is a compact monoid containing no idempotent apart from its identity. If $s\in S$, then $sS$ and $Ss$ are compact semigroups and so contain an idempotent which must be $1$ by assumption on $S$. It follows that $s$ is invertible. Thus $S$ is a group. In fact, it is easy to see that the inverse operation is then forced to be continuous and so $S$ is a compact group.

Answer 2

Topological semigroups, by A. B. Paalman-de Miranda, Mathematisch Centrum (1964):

Theorem 1.4.2. Let $S$ be a compact smgrp with $S^2=S$ and $S$ has a unique idempotent. Then S is a group.