# Commutative loop

From Groupprops

This article defines a property that can be evaluated for a loop.

View other properties of loops

ANALOGY: This is an analogue in loop of a property encountered in group. Specifically, it is a loop property analogous to the group property: abelian group

View other analogues of abelian group | View other analogues in loops of group properties (OR, View as a tabulated list)

## Contents

## Definition

A loop is termed a **commutative loop** or **abelian loop** if its binary operation is a commutative binary operation, i.e., it is a commutative magma under its multiplication.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Commutative Moufang loop | ||||

Abelian group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Commutative magma | ||||

Flexible loop |