Moufang implies diassociative

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., diassociative loop)
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Statement

Any Moufang loop is a diassociative loop.

Related facts

Similar facts

• Artin's theorem on alternative rings

Facts used

1. Moufang implies alternative: A Moufang loop is an alternative loop, i.e., it satisfies the left alternative law $x*(x*y)=(x*x)*y$ and the right alternative law $x*(y*y)=(x*y)*y$.
2. Moufang's theorem: This states that if $L$ is a Moufang loop, and $x,y,z$ are (not necessarily distinct) elements of $L$ such that $x*(y*z)=(x*y)*z$, then the subloop of $L$ generated by $x,y,z$ is a group, i.e., it is associative.

Proof

Given: A Moufang loop $L$. Elements $x,y\in L$ (not necessarily distinct).

To prove: The subloop of $L$ generated by $x,y$ is associative.

Proof: Setting $z=y$, we see that by fact (1), $x*(y*z)=(x*y)*z$ (using the right alternative law part). Hence, by fact (2), the subloop generated by $x,y,z$ is a group. But since $z=y$, this is the same as the subloop generated by $x$ and $y$, completing the proof.