# Right Inverse for All is Left Inverse

## Theorem

Let $\struct {S, \circ}$ be a semigroup with a right identity $e_R$ such that:

$\forall x \in S: \exists x_R: x \circ x_R = e_R$

That is, every element of $S$ has a right inverse with respect to the right identity.

Then $x_R \circ x = e_R$, that is, $x_R$ is also a left inverse with respect to the right identity.

## Proof

Let $y = x_R \circ x$. Then:

 $\displaystyle y \circ e_R$ $=$ $\displaystyle y \circ \paren {y \circ y_R}$ Definition of Right Inverse Element $\displaystyle$ $=$ $\displaystyle \paren {y \circ y} \circ y_R$ as $\circ$ is associative $\displaystyle$ $=$ $\displaystyle y \circ y_R$ Product of Semigroup Element with Right Inverse is Idempotent: $y = x_R \circ x$ $\displaystyle$ $=$ $\displaystyle e_R$ Definition of Right Inverse Element

So $x_R \circ x = e_R$, and $x_R$ behaves as a left inverse as well as a right inverse with respect to the right identity.

$\blacksquare$