Solution of Belousov's problem

Maks A. Akivis; Vladislav V. Goldberg

Discussiones Mathematicae - General Algebra and Applications, Tome 21 (2001), p. 93-103 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $f_{i} (x_{i}) / x_{i} \neq f_{j} (x_{j}) / x_{j}$ . This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.

Publié le : 2001-01-01

Zbl 1002.20047

EUDML-ID : urn:eudml:doc:287630

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Maks A. Akivis; Vladislav V. Goldberg. Solution of Belousov's problem. Discussiones Mathematicae - General Algebra and Applications, Tome 21 (2001) pp. 93-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1030/

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