Solution of Belousov's problem

Maks A. Akivis; Vladislav V. Goldberg

Discussiones Mathematicae - General Algebra and Applications (2001)

  • Volume: 21, Issue: 1, page 93-103
  • ISSN: 1509-9415

Abstract

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The authors prove that a local n-quasigroup defined by the equation x n + 1 = F ( x ₁ , . . . , x ₙ ) = ( f ₁ ( x ₁ ) + . . . + f ₙ ( x ₙ ) ) / ( x ₁ + . . . + x ₙ ) , 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.

How to cite

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Maks A. Akivis, and Vladislav V. Goldberg. "Solution of Belousov's problem." Discussiones Mathematicae - General Algebra and Applications 21.1 (2001): 93-103. <http://eudml.org/doc/287630>.

@article{MaksA2001,
abstract = {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.},
author = {Maks A. Akivis, Vladislav V. Goldberg},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {n-ary quasigroup; reducible; irreducible; differentiable quasigroups; reducible quasigroups; irreducible quasigroups; webs; -ary quasigroups},
language = {eng},
number = {1},
pages = {93-103},
title = {Solution of Belousov's problem},
url = {http://eudml.org/doc/287630},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Maks A. Akivis
AU - Vladislav V. Goldberg
TI - Solution of Belousov's problem
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2001
VL - 21
IS - 1
SP - 93
EP - 103
AB - 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.
LA - eng
KW - n-ary quasigroup; reducible; irreducible; differentiable quasigroups; reducible quasigroups; irreducible quasigroups; webs; -ary quasigroups
UR - http://eudml.org/doc/287630
ER -

References

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  1. [1] V.D. Belousov, n-ary quasigroups (Russian), Izdat. 'Shtiintsa', Kishinev 1972, 227 pp. 
  2. [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). Zbl0199.05201
  3. [3] W. Blaschke, Einführung in die Geometrie der Waben, Birkhäuser-Verlag, Basel-Stuttgart 1955, 108 pp. (Russian transl. GITTL, Moscow 1959), 144 pp. Zbl0068.36501
  4. [4] V.V. Borisenko, Irreducible n-quasigroups on finite sets of composite order (Russian), Mat. Issled., Vyp. 51 (1979), 38-42. 
  5. [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. Zbl0247.20080
  6. [6] M.M. Glukhov, Varieties of (i, j)-reducible n-quasigroups (Russian), Mat. Issled., Vyp. 39 (1976), 67-72. 
  7. [7] M.M. Glukhov, On the question of reducibility of principal parastrophies of n-quasigroups (Russian), Mat. Issled., Vyp. 113 (1990), 37-41. 
  8. [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. [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. [10] V.V. Goldberg, Theory of Multicodimensional (n+1)-Webs, Kluwer Academic Publishers, Dordrecht, 1988, xxii+466 pp. Zbl0668.53001
  11. [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. Zbl30.0326.01
  12. [12] V.V. Ryzhkov, Conjugate nets on multidimensional surfaces (Russian), Trudy Moscow. Mat. Obshch. 7 (1958). 

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