Moufang loop

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

Definition

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

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

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	|FULL LIST, MORE INFO