# 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