# Left Identity while exists Left Inverse for All is Identity

## Theorem

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

$\forall x \in S: \exists x_L: x_L \circ x = e_L$

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

Then $e_L$ is also a right identity, that is, is an identity.

## Proof

From Left Inverse for All is Right Inverse we have that $x \circ x_L = e_L$.

Then:

 $\displaystyle x \circ e_L$ $=$ $\displaystyle x \circ \paren {x_L \circ x}$ Definition of Left Inverse Element $\displaystyle$ $=$ $\displaystyle \left({x \circ x_L}\right) \circ x$ as $\circ$ is associative $\displaystyle$ $=$ $\displaystyle e_L \circ x$ $x_L$ is a Right Inverse $\displaystyle$ $=$ $\displaystyle x$ Definition of Left Identity

So $e_L$ behaves as a right identity as well as a left identity.

That is, by definition, $e_L$ is an identity element.

$\blacksquare$