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. 



Idempotents in compact semigroups
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. 


