| Abstract:
Moufang loops occur in several areas of mathematics:
- As multiplicative structures in a non-desarguesian plane
- As the multiplicative structure of the non-zero elements of the
Octonions over reals
- As the multiplicative structure of points on a cubic
hypersurface.
It was Nathan Mendelsohn who first introduced the idea of an
"extended triple system" to study the quasigroups arising in various
geometric situations including the class of cubic quasigroups (see
[2], [4]). Analogous to the case of abelian groups, there is a
canonical correspondence between commutative Moufang loops and cubic
quasigroups (e.g. see [1]). Motivated by these ideas, here we prove
an analog of Manin's theorem for cubic quasigroups. In the coming
months, we will explore the connections with Moufang loops and the
little Desargues Theorem.
References
- G. Gratzer and R. Padmanabhan
On idempotent, commutative, and nonassociative groupoids.
Proc. Amer. Math. Soc. 28 (1971) 75--80.
- D. Johnson and N.S. Mendelsohn
Extended triple systems. Aequationes Math. 8 (1972), 291--298.
- Yu. I. Manin, Cubic Forms : algebra, geometry, arithmetic.
American Elsevier Publishing Co., New York, 1974.
- N. S. Mendelsohn, R. Padmanabhan and Barry Wolk
Straight edge constructions on planar cubic curves.
C. R. Math. Rep. Acad. Sci. Canada 10 (1988), no. 2, 77--82.
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