Difference between revisions of "Jordan magma"

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(Definition)
 
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==Definition==
 
==Definition==
  
A [[magma]] <math>(S,*)</math> is termed a ''''Jordan magma''' if it satisfies the following two conditions:
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A [[magma]] <math>(S,*)</math> is termed a '''Jordan magma''' if it satisfies the following two conditions:
  
 
# Commutativity: <math>\! x * y = y * x \ \forall \ x,y \in S</math>.
 
# Commutativity: <math>\! x * y = y * x \ \forall \ x,y \in S</math>.
# Jordan's identity: <math>\! (x * y) * (x * x) = x * (y * (x * x))</math>.
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# Jordan's identity: <math>\! (x * y) * (x * x) = x * (y * (x * x)) \ \forall x,y \in S</math>.
  
 
==Relation with other properties==
 
==Relation with other properties==
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| [[Stronger than::Flexible magma]] || <math>x * (y * x) = (x * y) * x</math> || (via commutativity)|| || {{intermediate notions short|flexible magma|Jordan magma}}
 
| [[Stronger than::Flexible magma]] || <math>x * (y * x) = (x * y) * x</math> || (via commutativity)|| || {{intermediate notions short|flexible magma|Jordan magma}}
 
|-
 
|-
| [[Stronger than::Power-associative magma]] || powers are well-defined|| [[Jordan implies power-associative]] || || {{intermediate notions short|power-associative magma|Jordan magma}}
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| [[Stronger than::Magma in which cubes are well-defined]] || <math>x * (x * x) = (x * x) * x</math> || (via commutativity, flexibility) || || {{intermediate notions short|magma in which cubes are well-defined|Jordan magma}}
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|-
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| [[Stronger than::Magma in which cubes and fourth powers are well-defined]] || <math>x^3</math> well-defined, all parenthesizations of <math>x^4</math> also equal || || {{intermediate notions short|magma in which cubes and fourth powers are well-defined|Jordan magma}}
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|-
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| [[Stronger than::Magma in which powers up to the fifth are well-defined]] || <math>x^3, x^4, x^5</math> all well-defined ||  || {{intermediate notions short|magma in which powers up to the fifth are well-defined|Jordan magma}}
 
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Latest revision as of 01:57, 4 March 2010

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties

Definition

A magma (S,*) is termed a Jordan magma if it satisfies the following two conditions:

  1. Commutativity: \! x * y = y * x \ \forall \ x,y \in S.
  2. Jordan's identity: \! (x * y) * (x * x) = x * (y * (x * x)) \ \forall x,y \in S.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian semigroup
Abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Commutative magma any two elements commute (by definition) |FULL LIST, MORE INFO
Flexible magma x * (y * x) = (x * y) * x (via commutativity) Commutative magma|FULL LIST, MORE INFO
Magma in which cubes are well-defined x * (x * x) = (x * x) * x (via commutativity, flexibility) Commutative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Magma in which cubes and fourth powers are well-defined x^3 well-defined, all parenthesizations of x^4 also equal Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Magma in which powers up to the fifth are well-defined x^3, x^4, x^5 all well-defined |FULL LIST, MORE INFO