Generalized Bol Loop

A algebraic loop L is a generalized Bol loop if for all elements x, y, and z of L,


for some map alpha:L->L. As the name suggests, these are generalizations of Bol loops; in particular, a Bol loop is a generalized Bol loop with respect to the identity map 1:L->L.

One can show that there is an algebraic duality between generalized Bol loops and algebraic loops which satisfy the half-Bol identity (Adeniran and Solarin 1999).

See also

Algebraic Loop, Bol Loop, Half-Bol Identity, Moufang Loop

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. "Generalized Bol Loop." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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