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 $\exists x_L: x_L \circ x = e_L$, that is $x$ 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.

$\ds \paren {x \circ x_L} \circ \paren {x \circ x_L}$

$=$

$\ds x \circ \paren {x_L \circ x} \circ x_L$

$\ds $

$=$

$\ds x \circ e_L \circ x_L$

$\ds $

$=$

$\ds x \circ x_L$

Definition of Left Identity

$\blacksquare$