Solution of Belousov's problem

• Autorzy: Maks A. Akivis , Vladislav V. Goldberg
• Języki publikacji: EN

Abstrakty

EN

The authors prove that a local n-quasigroup defined by the equation
$x_{n+1} = F(x₁,...,xₙ) = (f₁(x₁) + ... + fₙ(xₙ))/(x₁ + ... + xₙ)$,
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.

Słowa kluczowe

EN

n-ary quasigroup; reducible; irreducible

Kategorie tematyczne

• 20N05: Loops, quasigroups

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2001
• Tom: 21
• Numer: 1
• Strony: 93-103

Daty

wydano

2001

otrzymano

2000-03-27

poprawiono

2000-10-09

Twórcy

autor

Maks A. Akivis

• Department of Mathematics, Jerusalem College of Technology - Mahon Lev, Havaad Haleumi St., P.O.B. 16031, Jerusalem 91160, Israel

autor

Vladislav V. Goldberg

• Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102

Bibliografia

• [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).

Identyfikatory

DOI

10.7151/dmgaa.1030