Diassociative loop

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This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

A diassociative loop is a loop with the property that the subloop generated by any subset of size at most two is a group, i.e., is associative.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group associative loop Moufang loop|FULL LIST, MORE INFO
Moufang loop loop satisfying a bunch of identities called Moufang identities Moufang's theorem |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
power-associative loop subloop generated by any element is associative |FULL LIST, MORE INFO
flexible loop satisfies the identity x * (y * x) = (x * y) * x for all x,y |FULL LIST, MORE INFO
alternative loop satisfes the left and right alternative laws |FULL LIST, MORE INFO
left alternative loop satisfies the left alternative law x * (x * y) = (x * x) * y for all x,y |FULL LIST, MORE INFO
right alternative loop satisfies the right alternative law x * (y * y) = (x * y) * y for all x,y |FULL LIST, MORE INFO