Left nuclear square loop

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This article defines a property that can be evaluated for a loop.
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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