Right-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 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