Moufang loop: Difference between revisions
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{{ | {{loop property}} | ||
{{variation of|group}} | |||
{{quick phrase|[[quick phrase::loop (identity, inverses, not necessarily associative) with some associativity-like conditions that come close to making it a group]]}} | |||
==Definition== | ==Definition== | ||
A '''Moufang loop''' is | ===In terms of Moufang's identities=== | ||
A '''Moufang loop''' is a [[loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities: | |||
# <math>\! z * (x * (z * y)) = ((z * x) * z) * y \ \forall \ x,y,z \in L</math> | # <math>\! z * (x * (z * y)) = ((z * x) * z) * y \ \forall \ x,y,z \in L</math> | ||
# <math>\! x * (z * (y * z)) = ((x * z) * y) * z \ \forall \ x,y,z \in L</math> | # <math>\! x * (z * (y * z)) = ((x * z) * y) * z \ \forall \ x,y,z \in L</math> | ||
# <math>\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L</math> | # <math>\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L</math> | ||
===In terms of Bol loops=== | |||
A '''Moufang loop''' is a [[loop]] that is both a [[defining ingredient::left Bol loop]] and a [[defining ingredient::right Bol loop]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Stronger than:: | | [[Stronger than::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]] || {{intermediate notions short|diassociative loop|Moufang loop}} | ||
|- | |||
| [[Stronger than::alternative loop]] || loop satisfying the left-alternative and right-alternative identities || [[Moufang implies alternative]] || [[alternative not implies Moufang]] || {{intermediate notions short|alternative loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::left alternative loop]] || loop satisfying the left alternative identity <math>x * (x * y) = (x * x) * y</math>|| || || {{intermediate notions short|left alternative loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::right alternative loop]] || loop satisfying the right alternative identity <math>x * (y * y) = (x * y) * y</math> || || || {{intermediate notions short|right alternative loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::flexible loop]] || loop satisfying the flexible law <math>x * (y * x) = (x * y) * x</math> || || || {{intermediate notions short|flexible loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::power-associative loop]] || loop in which the subloop generated by any element is a subgroup || || || {{intermediate notions short|power-associative loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::left Bol loop]] || satisfies the left Bol identity || || || {{intermediate notions short|left Bol loop|Moufang loop}} | |||
|- | |||
| [[Stronger than::right Bol loop]] || satisfies the right Bol identity || || || {{intermediate notions short|right Bol loop|Moufang loop}} | |||
|} | |} | ||
Latest revision as of 22:20, 24 June 2012
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
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 | |FULL LIST, MORE INFO |
| left alternative loop | loop satisfying the left alternative identity | |FULL LIST, MORE INFO | ||
| right alternative loop | loop satisfying the right alternative identity | |FULL LIST, MORE INFO | ||
| flexible loop | loop satisfying the flexible law | |FULL LIST, MORE INFO | ||
| power-associative loop | loop in which the subloop generated by any element is a subgroup | |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 |