Diassociative loop

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			|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