# Invertible Element of Associative Structure is Cancellable

## Theorem

Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is associative.

Let $\struct {S, \circ}$ have an identity element $e_S$.

An element of $\struct {S, \circ}$ which is invertible is also cancellable.

### Corollary

Let $\struct {S, \circ}$ be a monoid whose identity element is $e_S$.

An element of $\struct {S, \circ}$ which is invertible is also cancellable.

## Proof

Let $a \in S$ be invertible.

Suppose $a \circ x = a \circ y$.

Then:

 $\ds x$ $=$ $\ds e_S \circ x$ Definition of Identity Element $\ds$ $=$ $\ds \paren {a^{-1} \circ a} \circ x$ Definition of Inverse Element $\ds$ $=$ $\ds a^{-1} \circ \paren {a \circ x}$ Associativity of $\circ$ $\ds$ $=$ $\ds a^{-1} \circ \paren {a \circ y}$ by hypothesis $\ds$ $=$ $\ds \paren {a^{-1} \circ a} \circ y$ Associativity of $\circ$ $\ds$ $=$ $\ds e_S \circ y$ Definition of Inverse Element $\ds$ $=$ $\ds y$ Definition of Identity Element

A similar argument shows that $x \circ a = y \circ a \implies x = y$.

$\blacksquare$