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 |