Product of Semigroup Element with Right Inverse is Idempotent

Theorem
Let $\left({S, \circ}\right)$ be a semigroup with a right identity $e_R$.

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

Then:


 * $\left({x_R \circ x}\right) \circ \left({x_R \circ x}\right) = x_R \circ x$

... that is, $x_R \circ x$ is idempotent.

Also see

 * Product of Semigroup Element with Left Inverse is Idempotent