PDF

Discussiones Mathematicae General Algebra and Applications 21(1) (2001) 93-103
DOI: https://doi.org/10.7151/dmgaa.1030

SOLUTION OF BELOUSOV'S PROBLEM

Maks A. Akivis Department of Mathematics, Jerusalem College of Technology - Mahon Lev, Havaad Haleumi St., P.O.B. 16031, Jerusalem 91160, Israel e-mail: akivis@avoda.jct.ac.il

Vladislav V. Goldberg Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 e-mail vlgold@m.njit.edu

Abstract

The authors prove that a local n-quasigroup defined by the equation

where f_i (x_i), i, j = 1, …, n, are arbitrary functions, is irreducible if and only if any two functions f_i (x_i) and f_j (x_j), i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but [(f_i (x_i))/(x_i)] ≠ [(f_j(x_j))/(x_j)]. This gives a solution of Belousov's problem to construct examples of irreducible n-quasigroups for any n > 3.

Keywords: n-ary quasigroup, reducible, irreducible.

2000 Mathematics Subject Classification: Primary 20N05.

References

[1]	V.D. Belousov, n-ary quasigroups (Russian), Izdat. ``Shtiintsa'', Kishinev 1972, 227 pp.
[2]	V.D. Belousov, and M. D. Sandik, n-ary quasigroups and loops (Russian), Sibirsk. Mat. Zh. 7 (1966), no. 1, 31-54. (English transl. in: Siberian Math. J. 7 (1966), no. 1, 24-42).
[3]	W. Blaschke, Einführung in die Geometrie der Waben, Birkhäuser-Verlag, Basel-Stuttgart 1955, 108 pp. (Russian transl. GITTL, Moscow 1959, 144 pp.
[4]	V.V. Borisenko, Irreducible n-quasigroups on finite sets of composite order (Russian), Mat. Issled., Vyp. 51 (1979), 38-42.
[5]	B.R. Frenkin, Reducibility and uniform reducibility in certain classes of n-groupoids II (Russian), Mat. Issled., Vyp. 7 (1972), no. 1 (23), 150-162.
[6]	M.M. Glukhov, Varieties of (i, j)-reducible n-quasigroups (Russian), Mat. Issled., Vyp. 39 (1976), 67-72.
[7]	M.M. Glukhov, On the question of reducibility of principal parastrophies of n-quasigroups (Russian), Mat. Issled., Vyp. 113 (1990), 37-41.
[8]	V.V. Goldberg, The invariant characterization of certain closure conditions in ternary quasigroups (Russian), Sibirsk. Mat. Zh. 16 (1975), no. 1, 29-43. (English transl. in: Siberian Math. J. 16 (1975), no. 1, 23-34).
[9]	V.V. Goldberg, Reducible (n+1)-webs, group (n+1)-webs, and (2n+2)-hedral (n+1)-webs of multidimensional surfaces (Russian), Sibirsk. Mat. Zh. 17 (1976), no. 1, 44-57. (English transl. in: Siberian Math. J. 17 (1976), no. 1, 34-44).
[10]	V.V. Goldberg, Theory of Multicodimensional (n+1)-Webs, Kluwer Academic Publishers, Dordrecht, 1988, xxii+466 pp.
[11]	E. Goursat, Sur les équations du second ordre a n variables, analogues a l'équation de Monge-Ampere, Bull. Soc. Math. France 27 (1899), 1-34.
[12]	V.V. Ryzhkov, Conjugate nets on multidimensional surfaces (Russian), Trudy Moscow. Mat. Obshch. 7 (1958), 179-226.

Received 27 March 2000
Revised 9 October 2000