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# Algebraic Loop

A quasigroup with an identity element such that and for any in the quasigroup. All groups are loops.

In general, loops are considered to have very little in the way of algebraic structure and it is for that reason that many authors limit their investigation to loops which satisfy various other structural conditions. Common examples of such notions are the left- and right-Bol loop, the Moufang loop (which is both a left-Bol loop and a right-Bol loop simultaneously), and the generalized Bol loop.

The above definition of loop is purely algebraic and shouldn't be confused with other notions of loop, such as a closed curves, a multi-component knot or hitch, a graph loop, etc.

Bol Loop, Generalized Bol Loop, Group, Half-Bol Identity, Loop, Moufang Loop, Power Associative Algebra, Quasigroup

Portions of 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." Ann. St. Univ. "Al.I.Cuza", S. Mat. XLV, 99-102, 1999.Albert, A. A. (Ed.). Studies in Modern Algebra. Washington, DC: Math. Assoc. Amer., 1963.Moorhouse, G. E. "Bol Loops of Small Order." 2007. http://www.uwyo.edu/moorhouse/pub/bol/.Pedersen, J. "Loops." http://www.math.usf.edu/~eclark/algctlg/loops.html.

## Cite this as:

Stover, Christopher and Weisstein, Eric W. "Algebraic Loop." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlgebraicLoop.html