# Jordan magma

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 |