# Solution of Belousov's problem

@article{Akivis2000SolutionOB, title={Solution of Belousov's problem}, author={M. Akivis and V. V. Goldberg}, journal={arXiv: Group Theory}, year={2000} }

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 \neq 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…

## 8 Citations

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It is shown that all n-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z"5 or Z"7 are reducible for n>=4.

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