Bol Loop

The term Bol loop refers to either of two classes of algebraic loops satisfying the so-called Bol identities. In particular, a left Bol loop is an algebraic loop L which, for all x, y, and z in L, satisfies the left Bol relation


Similarly, L is a right Bol loop provided it satisfies the right Bol relation


An algebraic loop which is both a left and right Bol loop is called a Moufang loop.

Some sources use the term Bol loop to refer to a right Bol loop, whereas some reserve the term for algebraic loops that are Moufang.

Although (left and right) Bol loops have relatively weak structural properties, one can show that such structures L are power associative. Thus, given an algebraic loop L, the element x^n in L is well-defined for all elements x in L and all integers n in Z independent of which order the multiplications are performed.

See also

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

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