# Invertible Element of Associative Structure is Cancellable

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## Contents

## Theorem

Let $\left({S, \circ}\right)$ be an algebraic structure where $\circ$ is associative.

Let $\left({S, \circ}\right)$ have an identity element $e_S$.

An element of $\left({S, \circ}\right)$ which is invertible is also cancellable.

### Corollary

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

An element of $\left({S, \circ}\right)$ which is invertible is also cancellable.

## Proof

Let $a \in S$ be invertible.

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

Then:

\(\displaystyle x\) | \(=\) | \(\displaystyle e_S \circ x\) | Behaviour of Identity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({a^{-1} \circ a}\right) \circ x\) | Behaviour of Inverse | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a^{-1} \circ \left({a \circ x}\right)\) | Associativity of $\circ$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a^{-1} \circ \left({a \circ y}\right)\) | By Hypothesis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({a^{-1} \circ a}\right) \circ y\) | [Definition:Associative Operation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e_S \circ y\) | Behaviour of Inverse | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y\) | Behaviour of Identity |

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

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$: Theorem $7.1$