# 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 Alternative magma|FULL LIST, MORE INFO
Semigroup associativity holds universally Alternative magma, Diassociative 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 $x * (x * x) = (x * x) * x$ |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 $x * (x * y) = (x * x) * y$ Alternative magma, Diassociative magma|FULL LIST, MORE INFO Magma in which cubes are well-defined|FULL LIST, MORE INFO
Flexible magma $x * (y * x) = (x * y) * x$ Diassociative magma|FULL LIST, MORE INFO Magma in which cubes are well-defined|FULL LIST, MORE INFO