Commutative magma

From Groupprops
Revision as of 22:52, 2 March 2010 by Vipul (talk | contribs) (Created page with '{{magma property}} ==Definition== A magma <math>(S,*)</math> is termed a '''commutative magma''' (or sometimes '''abelian magma''') if it satisfies commutativity, i.e.,…')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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