# Invertible implies cancellative in monoid

## Statement

In a monoid (a set with an associative binary operation possessing a multiplicative identity element) the following are true:

## Proof

We'll give here the proof for left invertible and left cancellative. An analogous proof works for right invertible and right cancellative.

Given: A monoid $M$ with binary operation $*$, and identity element (also called neutral element) $e$. $a \in M$ has a left inverse $b$ (i.e. an element $b * a = e$)

To prove: $a$ is left-cancellative: whenever $c,d \in M$ are such that $a * c = a * d$, then $c =d$

$a * c = a * d$
Left-multiply both sides by $b$:
$b * (a * c) = b * (a * d)$
$(b * a) * c = (b * a) * d$
We now use that $b * a = e$ is the identity element, to conclude that $c = d$.