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Mat. Sb., 2016, Volume 207, Number 2, Pages 81–92 (Mi msb8447)  

This article is cited in 4 scientific papers (total in 4 papers)

The Post-Gluskin-Hosszú theorem for finite $n$-quasigroups and self-invariant families of permutations

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We study finite $n$-quasigroups $(n\geqslant3)$ with the following property of additional invertibility: if the quasigroup operation gives the same results on some two tuples of $n$ arguments with the same first components, then the tuples of the other $n-1$ components effect the same left translations. We prove an analogue of the Post-Gluskin-Hosszú theorem for such $n$-quasigroups. This has been proved previously, but only in the associative case. The theorem reduces the operation of the $n$-quasigroup to a group operation. The main tool used in the proof is a two-parameter self-invariant family of permutations on an arbitrary finite set. This is introduced and studied in the paper.
Bibliography: 13 titles.

Keywords: $n$-quasigroup, associativity, $n$-ary group, automorphism, Latin hypercube.

DOI: https://doi.org/10.4213/sm8447

Full text: PDF file (502 kB)
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English version:
Sbornik: Mathematics, 2016, 207:2, 226–237

Bibliographic databases:

UDC: 512.548.74
MSC: Primary 20N15; Secondary 20N05
Received: 11.11.2014 and 20.05.2015

Citation: F. M. Malyshev, “The Post-Gluskin-Hosszú theorem for finite $n$-quasigroups and self-invariant families of permutations”, Mat. Sb., 207:2 (2016), 81–92; Sb. Math., 207:2 (2016), 226–237

Citation in format AMSBIB
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  • https://doi.org/10.4213/sm8447
  • http://mi.mathnet.ru/eng/msb/v207/i2/p81

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Cheremushkin, “Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent $n$-ary operations”, Discrete Math. Appl., 29:5 (2019), 295–302  mathnet  crossref  crossref  isi  elib
    2. A. V. Cheremushkin, “Obobschenie teorem Gluskina–Khossu i Malysheva na sluchai cilno zavisimykh $n$-arnykh operatsii”, PDM. Prilozhenie, 2018, no. 11, 23–25  mathnet  crossref
    3. F. M. Malyshev, “Slabo obratimye $n$-kvazigruppy”, Chebyshevskii sb., 19:2 (2018), 304–318  mathnet  crossref  elib
    4. A. V. Cheremushkin, “Partially invertible strongly dependent $n$-ary operations”, Sb. Math., 211:2 (2020), 291–308  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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