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2001 | 21 | 1 | 93-103
Tytuł artykułu

Solution of Belousov's problem

Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Rocznik
Tom
21
Numer
1
Strony
93-103
Opis fizyczny
Daty
wydano
2001
otrzymano
2000-03-27
poprawiono
2000-10-09
Twórcy
  • Department of Mathematics, Jerusalem College of Technology - Mahon Lev, Havaad Haleumi St., P.O.B. 16031, Jerusalem 91160, Israel
  • 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).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1030
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