# Tour:Equality of left and right inverses

**This article adapts material from the main article:** equality of left and right inverses in monoid

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WHAT YOU NEED TO DO:

- Read, and understand, the statement below, and try proving it.
- Read the proof and make sure you understand it, as well as the significance of associativity.
PONDER (WILL BE EXPLORED LATER): What happens when we remove associativity? Can you cook up binary operations where left and right inverses exist but are no longer equal?

## Contents

## Statement

### Verbal statement

Suppose is the associative binary operation of a monoid, and is its neutral element (or identity element). If an element has both a left and a right inverse with respect to , then the left and right inverse are equal.

### Statement with symbols

Suppose is a monoid with binary operation and neutral element . If an element has a left inverse (i.e., )and a right inverse (i.e., ), then .

- Two-sided inverse is unique if it exists in monoid
- In a monoid, if an element has a left inverse, it can have at most one right inverse; moreover, if the right inverse exists, it must be equal to the left inverse, and is thus a two-sided inverse.
- In a monoid, if an element has a right inverse, it can have at most one left inverse; moreover, if the left inverse exists, it must be equal to the right inverse, and is thus a two-sided inverse.
- In a monoid, if an element has two distinct left inverses, it cannot have a right inverse, and hence cannot have a two-sided inverse.
- In a monoid, if an element has two distinct right inverses, it cannot have a left inverse, and hence cannot have a two-sided inverse.
- In a group, every element has a unique left inverse (same as its two-sided inverse) and a unique right inverse (same as its two-sided inverse).

## Proof

### Proof idea

The idea is to *pit* the left inverse of an element against its right inverse. Starting with an element , whose left inverse is and whose right inverse is , we need to form an expression that pits against , and can be simplified both to and to .

The only relation known between and is their relation with : is the neutral element and is the neutral element. To use both these facts, we construct the expression . The two ways of parenthesizing this expression allow us to simplify the expression in different ways.

The key idea here is that since and are related through , we *need* to put in between them in the expression. Then, we *need* associativity to interpret the expression in different ways and simplify to obtain the result.

### Formal proof

**Given**: A monoid with associative binary operation and neutral element . An element of with left inverse and right inverse .

**To prove**:

**Proof**: We consider two ways of associating the expression .

by associativity. The left side simplifies to while the right side simplifies to . Hence, .

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: Equality of left and right neutral element|UP: Introduction two (beginners)|NEXT: Equivalence of definitions of group