I found the following question in a test paper:
Suppose $G$ is a monoid or a semigroup. $a\in G$ and $a^2=a$. What can we say about $a$?
Monoids are associative and have an identity element. Semigroups are just associative.
I'm not sure what we can say about $a$ in this case other than that $a$ could be other things apart from the identity. Any idea if there's a definitive answer to this question?