Left nuclear square loop

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

Definition

An algebra loop $(L,*)$ is termed a left nuclear square loop if every square element, i.e., every element of the form $x*x$, is in the left nucleus. In other words, the following identity is satisfied for all $x,y,z\in L$:

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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
LC-loop
C-loop
Extra loop
Group

Incomparable properties

• Middle nuclear square loop
• Right nuclear square loop