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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

Contents

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