Invertible implies cancellative in monoid
- Any left invertible element (element having a left inverse) is left cancellative.
- Any right invertible element (element having a right inverse) is right cancellative.
- Any invertible element is cancellative.
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 with binary operation , and identity element (also called neutral element) . has a left inverse (i.e. an element )
To prove: is left-cancellative: whenever are such that , then
Proof: We start with:
Left-multiply both sides by :
We now use that is the identity element, to conclude that .