Moufang Loop

An algebraic loop L is a Moufang loop if all triples of elements x, y, and z in L satisfy the Moufang identities, i.e., if

1. z(x(zy))=((zx)z)y,

2. x(z(yz))=((xz)y)z,

3. (zx)(yz)=(z(xy))z, and

4. (zx)(yz)=z((xy)z).

One can show that an algebraic loop L which satisfies both the left and right Bol identities is Moufang.

See also

Algebraic Loop, Bol Loop, Generalized Bol Loop, Half-Bol Identity, Moufang Identities, Power Associative Algebra

This entry contributed by Christopher Stover

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Adeniran, J. O. and Solarin, A. R. T. "A Note on Generalized Bol Identity." An. Stiinţ. Univ. Al. I. Cuza Iasi. Mat. 45, 99-102, 1999.Moorhouse, G. E. "Bol Loops of Small Order." 2007.

Cite this as:

Stover, Christopher. "Moufang Loop." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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