# 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.

## Proof

 $\displaystyle \paren {x \circ x_L} \circ \paren {x \circ x_L}$ $=$ $\displaystyle x \circ \paren {x_L \circ x} \circ x_L$ as $\circ$ is associative $\displaystyle$ $=$ $\displaystyle x \circ e_L \circ x_L$ Definition of Left Inverse Element $\displaystyle$ $=$ $\displaystyle x \circ x_L$ Definition of Left Identity

$\blacksquare$