Left 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 left-alternative magma if it satisfies the following identity:

$x*(x*y)=(x*x)*y\mathrm{\forall }x,y\in S$.

Relation with other properties

Property obtained by the opposite operation

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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Alternative magma	both left and right-alternative			|FULL LIST, MORE INFO
Semigroup	associativity holds universally			|FULL LIST, MORE INFO
Diassociative magma	submagma generated by any two elements is associative			|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	every element commutes with its square			|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
Flexible magma	$x*(y*x)=(x*y)*x$			|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Right-alternative magma	$x*(y*y)=(x*y)*y$			|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
Power-associative magma	all powers are well-defined			|FULL LIST, MORE INFO	|FULL LIST, MORE INFO