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
Semigroup whole magma is associative Diassociative magma|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
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
Magma in which powers up to the fifth are well-defined x^3, x^4, x^5 well-defined for all x alternative implies powers up to the fifth are well-defined |FULL LIST, MORE INFO