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 an alternative magma if it is both a left-alternative magma and a right-alternative magma, i.e., it satisfies the following two identities:

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

Relation with other properties

Property obtained by the opposite operation

Suppose $(S,*)$ is a magma and we define $\cdot$ on $S$ as $a \cdot b := b * a$. Then, $(S,*)$ is an alternative magma if and only if $(S,\cdot)$ is an alternative magma.

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Diassociative magma submagma generated by any two elements is associative |FULL LIST, MORE INFO
Left-alternative magma $x * (x * y) = (x * x) * y$ |FULL LIST, MORE INFO
Right-alternative magma $x * (y * y) = (x * y) * y$ |FULL LIST, MORE INFO
Magma in which cubes are well-defined $x * (x * x) = (x * x) * x$ Left alternative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined, Right-alternative magma|FULL LIST, MORE INFO