Inverse in Monoid is Unique

Theorem
Let $\left({S, \circ}\right)$ be a monoid.

Then an element $x \in S$ can have at most one inverse for $\circ$.

Proof
Let $e$ be the identity element of $\left({S, \circ}\right)$.

Suppose $x \in S$ has two inverses: $y$ and $z$.

Then:

Similarly:

So whichever way round you do it, $y = z$ and the inverse of $x$ is unique.

Also see

 * Inverse not always Unique for Non-Associative Operation
 * Identity is Unique
 * Inverse in Group is Unique