# Commutative magma

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

$x * y = y * x \ \forall \ x,y \in S$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian 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 $x * (y * x) = (x * y) * x$ for all $x,y$ |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