LC implies middle nuclear square

This article gives the statement and possibly, proof, of an implication relation between two algebra loop properties. That is, it states that every algebra loop satisfying the first algebra loop property (i.e., LC-loop) must also satisfy the second algebra loop property (i.e., middle nuclear square loop)
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Statement

Any LC-loop is a middle nuclear square loop.

Definitions used

LC-loop

Further information: LC-loop

An algebra loop $(L,*)$ is termed a LC-loop if it satisfies the following identity for all $x,y,z\in L$:

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

Middle nuclear square loop

Further information: Middle nuclear square loop

An algebra loop $(L,*)$ is termed a middle nuclear square loop if it satisfies the following identity for all $x,y,z\in L$:

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

Related facts

• LC implies left alternative
• LC implies left nuclear square

Proof

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