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