Moufang implies alternative

This article gives the statement and possibly, proof, of an implication relation between two [[{{{context space}}} property|{{{context space}}} properties]]. That is, it states that every {{{context space}}} satisfying the first {{{context space}}} property (i.e., Moufang loop) must also satisfy the second {{{context space}}} property (i.e., alternative loop)
[[:Category:{{{context space}}} property implications|View all {{{context space}}} property implications]] | [[:Category:{{{context space}}} property non-implications|View all {{{context space}}} property non-implications]]
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[[Category: {{{context space}}} property implications]]

Statement

Any Moufang loop is an alternative loop.

Related facts

Stronger facts

• Moufang implies diassociative

Applications

• Moufang implies diassociative

Definitions used

Moufang loop

Further information: Moufang loop

A Moufang loop is a loop $L$ satisfying the following identities for all $x,y,z\in L$ (where two or more of the $x,y,z$ could possibly be equal):

1. $\!z*(x*(z*y))=((z*x)*z)*y\ \forall \ x,y,z\in L$
2. $\!x*(z*(y*z))=((x*z)*y)*z\ \forall \ x,y,z\in L$
3. $\!(z*x)*(y*z)=(z*(x*y))*z\ \forall \ x,y,z\in L$

Alternative loop

Further information: alternative loop

An alternative loop is a loop $L$ satisfying the following two identities for all $a,b\in L$ (where $a,b$ may be equal or distinct):

• $a*(a*b)=(a*a)*b$
• $a*(b*b)=(a*b)*b$

Proof

Given: A Moufang loop $L$ with identity element $e$.

To prove: For all $a,b\in L$ (possibly equal), we have $a*(a*b)=(a*a)*b$ (left alternative law) and $a*(b*b)=(a*b)*b$ (right alternative law).

Proof: For the left alternative law, set $x=e$, $y=b$, and $z=a$ in Moufang's identity (1) given above.

For the right alternative law, set $x=a$, $y=e$, and $z=b$ in Moufang's identity (2) given above.