Right-alternative 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 right-alternative magma if it satisfies the following identity:

$\!x*(y*y)=(x*y)*y\ \forall \ x,y\in S$

Relation with other properties

Property obtained by the opposite operation

If we consider a magma $(S,*)$ and now define $\cdot$ on $S$ by $a\cdot b:=b*a$, then $(S,*)$ is a right-alternative magma if and only if $(S,\cdot )$ is a left-alternative magma.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Alternative magma	both right- and left-alternative
Diassociative magma	submagma generated by any two elements is associative			|FULL LIST, MORE INFO
Semigroup	associativity holds universally			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Magma in which cubes are well-defined	$x*(x*x)=(x*x)*x$			|FULL LIST, MORE INFO

Incomparable properties

Property	Meaning	Proof of one non-implication	Proof of other non-implication	Notions stronger than both	Notions weaker than both
Left-alternative magma	$x*(x*y)=(x*x)*y$			|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Flexible magma	$x*(y*x)=(x*y)*x$			|FULL LIST, MORE INFO	|FULL LIST, MORE INFO