An element e of a monoid M is called an idempotent if e^2 = e. If M is finite, show that some positive power of every element is an idempotent.
Prove by induction:
e is idempotent, then e^2=e (1)
Assuming when the positive power of e is n=k, then (e^k) ...
Idempotence of powers of elements in a monoid is demonstrated by mathematical induction.