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