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Solution of Belousov's problem

Maks A. Akivis;  Vladislav V. Goldberg — 2001

Discussiones Mathematicae - General Algebra and Applications

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.

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