Geodesic
An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations
Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 23-25 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $x_i\in X$, $i=1,\dots,n$.
Keywords: $n$-ary group, $n$-ary semigroup, strong dependent operation, weak invertible operation. 
@article{PDMA_2018_11_a6,
     author = {A. V. Cheremushkin},
     title = {An extension of {Gluskin{\textendash}Hoszu's} and {Malyshev's} theorems to strong dependent $n$-ary operations},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {23--25},
     year = {2018},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2018_11_a6/}
}
TY  - JOUR
AU  - A. V. Cheremushkin
TI  - An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2018
SP  - 23
EP  - 25
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/PDMA_2018_11_a6/
LA  - ru
ID  - PDMA_2018_11_a6
ER  - 
%0 Journal Article
%A A. V. Cheremushkin
%T An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2018
%P 23-25
%N 11
%U http://geodesic.mathdoc.fr/item/PDMA_2018_11_a6/
%G ru
%F PDMA_2018_11_a6
A. V. Cheremushkin. An extension of Gluskin–Hoszu's and Malyshev's theorems to strong dependent $n$-ary operations. Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 23-25. http://geodesic.mathdoc.fr/item/PDMA_2018_11_a6/

[1] Galmak A. M., Vorobev G. N., “O teoreme Posta–Gluskina–Khossu”, Problemy fiziki, matematiki i tekhniki, 2013, no. 1(14), 55–60

[2] Malyshev F. M., “O teoreme Posta–Gluskina–Khossu dlya konechnykh kvazigrupp i samoinvariantnye semeistva podstanovok”, Matematicheskii sbornik, 207:2 (2016), 81–92 | DOI | MR | Zbl

[3] Malyshev F. M., “Teorema Posta–Gluskina–Khossu dlya $n$-kvazigrupp”, Issledovaniya po algebre, teorii chisel, funktsionalnomu analizu i smezhnym voprosam, Mezhvuz. sb. nauchn. tr., v. 8, Izd-vo Sarat. un-ta, Saratov, 2016, 59–62

[4] Sokhatskii F. N., “Obobschenie dvukh teorem Belousova dlya silno zavisimykh funktsii $k$-znachnoi logiki”, Issledovaniya po teorii binarnykh i $n$-arnykh kvazigrupp, Matematicheskie issledovaniya, 85, Shtiintsa, Kishinev, 1985, 105–115 | MR