# Left alternative 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 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
Alternative magma both left and right-alternative |FULL LIST, MORE INFO
Semigroup associativity holds universally Alternative magma, Diassociative magma|FULL LIST, MORE INFO
Diassociative magma submagma generated by any two elements is associative Alternative magma|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 $x * (y * x) = (x * y) * x$ 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
Power-associative magma all powers are well-defined Diassociative magma|FULL LIST, MORE INFO Magma in which cubes are well-defined|FULL LIST, MORE INFO