An extension of gluskin - hoszu's and maly-shev's theorems to strong dependent n-ary operations

The report presents an extension of Malyshev theorem for n-ary quasigroups with a right or left weak inverse property to the case of strong dependent n-ary operations on a finite set. The main result is the following theorem. Let n ^ 3 and a strong dependent n-ary function f on a finite set X be such that f (x₁,...,x_n) = g₁(x, h(y, z)) = g₂(h(x, y), z), for all (x₁,... ,x_n) = (x,y,z) E X^г x X^n-i x X* and some g₁,g₂, h. Then there exist a permutation a, a monoid "*"on X and an automorphism 9 of "*" such that a(/(x1,..., x_ra)) = X1 * 9(x2) * 9²(хз) * ... * 9^n-1(x„), for all xj G X, i = 1,...,n. As a corollary, the following new proof of Gluskin - Hosszu theorem for strong dependent n-ary semigroups is obtained: if a strong dependent n-ary operation [x₁,..., x_n] admits an identity [[x₁,..., x_n], x_n+1,..., x_2n-1] = = [x₁, [x₂,... ,x_n+1],x_n+2,... ,x_2n-1], then there exist a monoid "*" on X and an automorphism 9 of "*" such that 9^n-1(x) = a * x * a^-1, a G X, 9(a) = a, and [x₁,... ,x_n] = = x₁ * 9(x₂) * 9²(x₃) * ... * 9^n-2(x_n-1) * a * x_n for all xj G X, i = 1,..., n.

Keywords

Authors

References