Moufang loop

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

Definition

In terms of Moufang's identities

A Moufang loop is a 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 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 $x * (x * y) = (x * x) * y$			Alternative loop, Diassociative loop|FULL LIST, MORE INFO
right alternative loop	loop satisfying the right alternative identity $x * (y * y) = (x * y) * y$			Diassociative loop|FULL LIST, MORE INFO
flexible loop	loop satisfying the flexible law $x * (y * x) = (x * y) * x$			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