Product of Commuting Idempotent Elements is Idempotent

Theorem

Let $\struct {S, \circ}$ be a semigroup.

Let $a, b \in S$ be idempotent elements of $S$.

Let $a$ and $b$ commute:

$a \circ b = b \circ a$

Then $a \circ b$ is idempotent.

Proof

 $\displaystyle \paren {a \circ b} \circ \paren {a \circ b}$ $=$ $\displaystyle \paren {a \circ \paren {b \circ a} } \circ b$ $\circ$ is associative by definition of semigroup $\displaystyle$ $=$ $\displaystyle \paren {a \circ \paren {a \circ b} } \circ b$ $a$ and $b$ commute $\displaystyle$ $=$ $\displaystyle \paren {a \circ a} \circ \paren {b \circ b}$ $\circ$ is associative $\displaystyle$ $=$ $\displaystyle a \circ b$ $a$ and $b$ are idempotent by the premise

Thus $a \circ b$ is idempotent.

$\blacksquare$