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Half-Bol Identity


In the study of non-associative algebra, there are at least two different notions of what the half-Bol identity is. Throughout, let L be an algebraic loop and let x, y, and z be elements of L.

Some authors use the term half-Bol to refer to the identity

 ((xy)z)y^alpha=x((yz)y^alpha)

for alpha in Z an integer. In this context, there is a strong algebraic duality between algebraic loops L which satisfy the above identity and those which are generalized Bol loops (Adeniran and Solarin 1999).

On the other hand, at least one author use the phrase half-Bol loop to refer to an algebraic loop L for which one can find a mapping tau:L->L such that

 ((xy)z)y^alpha=x((yz)y^alpha).

In this context, there is a considerable amount of variability as the mapping tau may or may not be assumed nonzero and may also be assumed to satisfy various other conditions as well (Boerner and Kallaher 1982).


See also

Algebraic Loop, Bol Loop, Generalized Bol Loop, Moufang Loop

This entry contributed by Christopher Stover

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References

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.Boerner, V. and Kallaher, M. J. "Half-Bol Quasi-Fields." J. Geom. 18, 185-193, 1982.Moorhouse, G. E. "Bol Loops of Small Order." 2007. http://www.uwyo.edu/moorhouse/pub/bol/.

Cite this as:

Stover, Christopher. "Half-Bol Identity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Half-BolIdentity.html

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