# Moufang loop

From Groupprops

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

View other properties of loops

This is a variation of group|Find other variations of group | Read a survey article on varying group

QUICK PHRASES: loop (identity, inverses, not necessarily associative) with some associativity-like conditions that come close to making it a group

## Contents

## Definition

### In terms of Moufang's identities

A **Moufang loop** is a loop with multiplication satisfying the following three identities:

### In terms of Bol loops

A **Moufang loop** is a loop that is both a left Bol loop and a right Bol loop.

## Relation with other properties

### Stronger properties

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

Group | an associative loop (see nonempty associative quasigroup equals group) | |||

Finite Moufang loop |

### Weaker properties

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

diassociative loop | loop in which the subloop generated by any subset of size at most two is a group | Moufang implies diassociative | diassociative not implies Moufang | |FULL LIST, MORE INFO |

alternative loop | loop satisfying the left-alternative and right-alternative identities | Moufang implies alternative | alternative not implies Moufang | Diassociative loop|FULL LIST, MORE INFO |

left alternative loop | loop satisfying the left alternative identity | Alternative loop, Diassociative loop|FULL LIST, MORE INFO | ||

right alternative loop | loop satisfying the right alternative identity | Diassociative loop|FULL LIST, MORE INFO | ||

flexible loop | loop satisfying the flexible law | Diassociative loop|FULL LIST, MORE INFO | ||

power-associative loop | loop in which the subloop generated by any element is a subgroup | Diassociative loop|FULL LIST, MORE INFO | ||

left Bol loop | satisfies the left Bol identity | |FULL LIST, MORE INFO | ||

right Bol loop | satisfies the right Bol identity | |FULL LIST, MORE INFO |