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

A. V. Cheremushkin

Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 23-25

Résumé

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 $f(x_1,\dots,x_n)=g_1(\bar x,h(\bar y,\bar z))=g_2(h(\bar x,\bar y),\bar z)$, for all $(x_1,\dots,x_n)=(\bar x,\bar y,\bar z)\in X^i\times X^{n-i}\times X^i$ 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 $$ \sigma(f(x_1,\dots,x_n))=x_1\ast\theta(x_2)\ast\theta^2(x_3)\ast\dots\ast\theta^{n-1}(x_n), $$ for all $x_i\in X$, $i=1,\dots,n$. 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 $[x_1,\dots,x_n]$ admits an identity $[[x_1,\dots,x_n],x_{n+1},\dots,x_{2n-1}]=[x_1,[x_2,\dots,x_{n+1}],x_{n+2},\dots,x_{2n-1}]$, 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 $[x_1,\dots,x_n]=x_1\ast\theta(x_2)\ast\theta^2(x_3)\ast\dots\ast\theta^{n-2}(x_{n-1})\ast a\ast x_n$ for all $x_i\in X$, $i=1,\dots,n$.

Détail

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