Moufang loop

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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