Question

prove that every finite semigroup has an idempotent element

Answer 1

Let (G,⋅)(G,⋅) be a finite semigroup. Is there any a∈Ga∈G such that:

a2=aa2=a

It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof?

Theorem 2.2.1. [R. Ellis] Let SS be a compact right topological semigroup. Then there exists an idempotent in it.

This theorem is also known as Ellis–Numakura lemma.