Product of Semigroup Element with Left Inverse is Idempotent

Theorem
Let $\struct {S, \circ}$ be a semigroup with a left identity $e_L$.

Let $x \in S$ such that $\exists x_L: x_L \circ x = e_L$, i.e. $x$ has a left inverse with respect to the left identity.

Then:


 * $\paren {x \circ x_L} \circ \paren {x \circ x_L} = x \circ x_L$

That is, $x \circ x_L$ is idempotent.

Also see

 * Product of Semigroup Element with Right Inverse is Idempotent