Jordan 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

Definition

A magma (S,*) is termed a 'Jordan magma if it satisfies the following two conditions:

  1. Commutativity: \! x * y = y * x \ \forall \ x,y \in S.
  2. Jordan's identity: \! (x * y) * (x * x) = x * (y * (x * x)).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian semigroup
Abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Commutative magma any two elements commute (by definition) |FULL LIST, MORE INFO
Flexible magma x * (y * x) = (x * y) * x (via commutativity) Commutative magma|FULL LIST, MORE INFO
Power-associative magma powers are well-defined Jordan implies power-associative |FULL LIST, MORE INFO