Left alternative loop

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

Definition

Symbol-free definition

A loop is termed a left-alternative loop if it is a Left-alternative magma (?).

Definition with symbols

An algebra loop $(L,*)$ is termed a left-alternative loop if the following identity is satisfied for all $x,y\in L$:

$\!x*(x*y)=(x*x)*y$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left Bol loop		left Bol implies left alternative		|FULL LIST, MORE INFO
Alternative loop				|FULL LIST, MORE INFO
Left Bruck loop				|FULL LIST, MORE INFO
Left power-alternative loop				|FULL LIST, MORE INFO