Moufang implies alternative
This article gives the statement and possibly, proof, of an implication relation between two loop properties. That is, it states that every loop satisfying the first loop property (i.e., Moufang loop) must also satisfy the second loop property (i.e., alternative loop)
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Further information: Moufang loop
A Moufang loop is a loop satisfying the following identities for all (where two or more of the could possibly be equal):
Further information: alternative loop
An alternative loop is a loop satisfying the following two identities for all (where may be equal or distinct):
Given: A Moufang loop with identity element .
To prove: For all (possibly equal), we have (left alternative law) and (right alternative law).
Proof: For the left alternative law, set , , and in Moufang's identity (1) given above.
For the right alternative law, set , , and in Moufang's identity (2) given above.