Difference between revisions of "Moufang loop"

From Groupprops
Jump to: navigation, search
(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…')
 
 
(4 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{algebra loop property}}
+
{{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 an [[algebra loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities:
+
===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==
Line 26: Line 34:
 
! 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::Alternative loop]] || algebra 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::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 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