Moufang implies alternative

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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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Statement

Any Moufang loop is an alternative loop.

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Applications

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.