# Inverse in Monoid is Unique

## Theorem

Let $\struct {S, \circ}$ 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 $\struct {S, \circ}$.

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

Then:

 $\displaystyle y$ $=$ $\displaystyle y \circ e$ Definition of Identity Element $\displaystyle$ $=$ $\displaystyle y \circ \paren {x \circ z}$ Definition of Inverse Element $\displaystyle$ $=$ $\displaystyle \paren {y \circ x} \circ z$ Definition of Associative Operation $\displaystyle$ $=$ $\displaystyle e \circ z$ Definition of Inverse Element $\displaystyle$ $=$ $\displaystyle z$ Definition of Identity Element

Similarly:

 $\displaystyle y$ $=$ $\displaystyle e \circ y$ Definition of Identity Element $\displaystyle$ $=$ $\displaystyle \paren {z \circ x} \circ y$ Definition of Inverse Element $\displaystyle$ $=$ $\displaystyle z \circ \paren {x \circ y}$ Definition of Associative Operation $\displaystyle$ $=$ $\displaystyle z \circ e$ Definition of Inverse Element $\displaystyle$ $=$ $\displaystyle z$ Definition of Identity Element

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

$\blacksquare$