# Difference between revisions of "Jordan magma"

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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:: | + | | [[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}} |

+ | |- | ||

+ | | [[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}} | ||

+ | |- | ||

+ | | [[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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## Revision as of 01:13, 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

## Contents

## Definition

A magma is termed a **Jordan magma** if it satisfies the following two conditions:

- Commutativity: .
- Jordan's identity: .

## 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 | (via commutativity) | Commutative magma|FULL LIST, MORE INFO | ||

Magma in which cubes are well-defined | (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 | well-defined, all parenthesizations of 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 | all well-defined | |FULL LIST, MORE INFO |