What can we conclude about $a$ if $a^2=a$ in a monoid or semigroup?

Question

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?

Answer 1

If the test is for an introductory abstract algebra course, I say $a$ is an idempotent element of $G$.

Answer 2

Several things can be said on $a$:

1. $a$ is idempotent,
2. $a$ is regular,
3. the subsemigroup generated by $a$ is trivial,
4. the $\mathcal{H}$-class of $a$ is a group.

Make your choice!

Answer 3

$$\{1,a,a^2,a^3,...\} = \{1,a,a,aa^2,...\} = \{1,a,a,a^2,...\} = \{1,a,a,a,...\} = \{1,a\}$$