# Commutative magma

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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 **commutative magma** (or sometimes **abelian magma**) if it satisfies commutativity, i.e., the following holds:

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## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Abelian group | Commutative loop, Jordan magma|FULL LIST, MORE INFO | |||

Abelian monoid | commutative, associative, has identity element | |FULL LIST, MORE INFO | ||

Abelian semigroup | commutative and associative | Jordan magma|FULL LIST, MORE INFO | ||

Jordan magma | commutative, also satisfies Jordan's identity | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Flexible magma | for all | |FULL LIST, MORE INFO | ||

Magma in which cubes are well-defined | every element commutes with its square | Flexible magma, Magma in which cubes are well-defined and every element commutes with its cube|FULL LIST, MORE INFO |