Difference between revisions of "Moufang loop"

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(Created page with '{{algebra loop property}} ==Definition== A '''Moufang loop''' is an algebra loop <math>L</math> with multiplication <math>*</math> satisfying the following three identities…')
 
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==Definition==
 
==Definition==
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===In terms of Moufang's identities===
  
 
A '''Moufang loop''' is an [[algebra loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities:
 
A '''Moufang loop''' is an [[algebra loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities:
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# <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>
  
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===In terms of Bol loops===
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A '''Moufang loop''' is an [[algebra 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==
  

Revision as of 23:18, 5 March 2010

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

Definition

In terms of Moufang's identities

A Moufang loop is an algebra 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 an algebra 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
Alternative loop algebra loop satisfying the left-alternative and right-alternative identities Moufang implies alternative alternative not implies Moufang Diassociative loop|FULL LIST, MORE INFO