# Equality of left and right nil element

From Groupprops

This article gives a statement (possibly with proof) of how, if a left-based construction and a right-based construction both exist, they must be equal.

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## Contents

## Statement

Suppose is a magma (a set with a binary operation ). Suppose is a left nil element (i.e., for all ) and is a right nil element for (i.e., for all ). Then, .

## Proof

### Proof idea

A left nil element can be thought of as an element that *dominates* the product when placed on the left, and a right nil element is an element that *dominates* the product when placed on the right. To show that these are equal, *we need to pit the left and right nil elements against each other.* Since both of them must dominate, they must both be equal.

### Formal proof

**Given**: A magma with binary operation , a left nil element for , a right nil element for .

**To prove**:

**Proof**: Consider the product . Since is a left nil element, . Since is a right nil element, . Thus, .