# Difference between revisions of "Moufang loop"

This article defines a property that can be evaluated for a loop.
View other properties of loops

## Definition

### In terms of Moufang's identities

A Moufang loop is an algebra loop $L$ with multiplication $*$ satisfying the following three identities:

1. $\! z * (x * (z * y)) = ((z * x) * z) * y \ \forall \ x,y,z \in L$
2. $\! x * (z * (y * z)) = ((x * z) * y) * z \ \forall \ x,y,z \in L$
3. $\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L$

### In terms of Bol loops

A Moufang loop is an algebra 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
Alternative loop algebra loop satisfying the left-alternative and right-alternative identities Moufang implies alternative alternative not implies Moufang Diassociative loop|FULL LIST, MORE INFO