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 Prikl. Diskr. Mat. Suppl., 2018, Issue 11, Pages 23–25 (Mi pdma377)

Theoretical Foundations of Applied Discrete Mathematics

An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations

A. V. Cheremushkin

Research Institute "Kvant", Moscow

Abstract: 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 $n\ge3$ and a strong dependent $n$-ary function $f$ on a finite set $X$ be such that $f(x_1,…,x_n)=g_1(\bar x,h(\bar y,\bar z))=g_2(h(\bar x,\bar y),\bar z)$, for all $(x_1,…,x_n)=(\bar x,\bar y,\bar z)\in X^i\times X^{n-i}\times X^i$ and some $g_1,g_2,h$. Then there exist a permutation $\sigma$, a monoid “$\ast$”on $X$ and an automorphism $\theta$ of “$\ast$” such that
$$\sigma(f(x_1,…,x_n))=x_1\ast\theta(x_2)\ast\theta^2(x_3)\ast…\ast\theta^{n-1}(x_n),$$
for all $x_i\in X$, $i=1,…,n$. As a corollary, the following new proof of Gluskin–Hosszú theorem for strong dependent $n$-ary semigroups is obtained: if a strong dependent $n$-ary operation $[x_1,…,x_n]$ admits an identity $[[x_1,…,x_n],x_{n+1},…,x_{2n-1}]=[x_1,[x_2,…,x_{n+1}],x_{n+2},…,x_{2n-1}]$, then there exist a monoid “$\ast$” on $X$ and an automorphism $\theta$ of “$\ast$” such that $\theta^{n-1}(x)=a\ast x\ast a^{-1}$, $a\in X$, $\theta(a)=a$, and $[x_1,…,x_n]=x_1\ast\theta(x_2)\ast\theta^2(x_3)\ast…\ast\theta^{n-2}(x_{n-1})\ast a\ast x_n$ for all $x_i\in X$, $i=1,…,n$.

Keywords: $n$-ary group, $n$-ary semigroup, strong dependent operation, weak invertible operation.

DOI: https://doi.org/10.17223/2226308X/11/7

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UDC: 519.719.1

Citation: A. V. Cheremushkin, “An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations”, Prikl. Diskr. Mat. Suppl., 2018, no. 11, 23–25

Citation in format AMSBIB
\Bibitem{Che18} \by A.~V.~Cheremushkin \paper An extension of Gluskin--Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations \jour Prikl. Diskr. Mat. Suppl. \yr 2018 \issue 11 \pages 23--25 \mathnet{http://mi.mathnet.ru/pdma377} \crossref{https://doi.org/10.17223/2226308X/11/7} \elib{https://elibrary.ru/item.asp?id=35557590}