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:

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

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
Flexible magma $x * (y * x) = (x * y) * x$ Diassociative magma, Left Bol magma with neutral element|FULL LIST, MORE INFO Magma in which cubes are well-defined|FULL LIST, MORE INFO
Right-alternative magma $x * (y * y) = (x * y) * y$ Alternative magma, Diassociative magma|FULL LIST, MORE INFO Magma in which cubes are well-defined|FULL LIST, MORE INFO