Jordan 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 Jordan magma if it satisfies the following two conditions:

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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Abelian 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)		|FULL LIST, MORE INFO
Magma in which cubes are well-defined	$x*(x*x)=(x*x)*x$	(via commutativity, flexibility)		|FULL LIST, MORE INFO
Magma in which cubes and fourth powers are well-defined	$x^{3}$ well-defined, all parenthesizations of $x^{4}$ also equal		|FULL LIST, MORE INFO
Magma in which powers up to the fifth are well-defined	$x^{3},x^{4},x^{5}$ all well-defined		|FULL LIST, MORE INFO