Left Bol implies left alternative

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., left Bol loop) must also satisfy the second algebra loop property (i.e., left alternative loop)
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Statement

Statement for algebra loops

Any algebra loop that is a left Bol loop is also a left alternative loop.

Statement for magmas

A magma that is a Left Bol magma (?) and has a Neutral element (?) must be a Left alternative magma (?).

Definitions used

Left Bol magma and loop

Further information: left Bol magma, left Bol loop

A magma $(L,*)$ is termed a left Bol magma if the following is satisfied for all $x,y,z\in L$:

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

A left Bol loop is a left Bol magma that is also an algebra loop.

Neutral element

A neutral element in a magma $(L,*)$ is an element $e$ such that $x*e=e*x=x$ for all $x\in L$. Note that any algebra loop has a neutral element by definition.

Left alternative magma and loop

Further information: left alternative magma, left alternative loop

A magma $(L,*)$ is termed a left alternative magma if the following is satisfied for all $x,y\in L$:

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

A left alternative loop is an algebra loop that is also a left alternative magma.

Proof

We prove the statement for magmas. Since an algebra loop has a neutral element by definition, the statement for algebra loops is an immediate consequence.

Given: A magma $(L,*)$ having a neutral element $e$, (i.e., $x*e=e*x=x$ for all $x\in L$) and satisfying the left Bol identity for all $x,y,z\in L$:

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

To prove: $L$ is left alternative, i.e., $x*(x*y)=(x*x)*y\mathrm{\forall }x,y\in L$

Proof: Set $y=e$ in the left Bol identity. We get:

$x*(e*(x*z))=(x*(e*x))*z$

This simplifies to:

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

Since $x,z$ are universally quantified, we can replace the dummy variable $z$ by the dummy variable $y$, and obtain:

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