# Tour:Invertible implies cancellative in monoid

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**This article adapts material from the main article:** invertible implies cancellative in monoid

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WHAT YOU NEED TO DO:

- Understand the statement below; try proving it yourself
- Understand the proof, and the crucial way in which it relies on associativity

PONDER:

- What happens if we remove the assumption of associativity? Can you cook up a magma (set with binary operation) having a neutral element, where an element has a left inverse but is not cancellative?

## Statement

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

- 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.

## 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 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 :

Use associativity:

We now use that is the identity element, to conclude that .

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Equivalence of definitions of group|UP: Introduction two (beginners)|NEXT: Equivalence of definitions of subgroup