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Chebyshevskii Sb., 2018, Volume 19, Issue 2, Pages 304–318 (Mi cheb656)  

This article is cited in 1 scientific paper (total in 1 paper)

Weakly invertible $ n $-quasigroups

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: We study the $ n $-quasigroups $ (n \geqslant3) $ with the following property weak invertibility. If on any two sets of $ n $ arguments with the equal initials, equal ends, but with different middle parts (of the same length), the result of the operation is the same, then for any identical beginnings (of a other length), with the previous middle parts and for any identical ends (the corresponding length), the result of the operation will be the same. For such $ n $-quasigroups An analog of the Post-Gluskin-Hoss theorem is proved, which reduces the operation of an $ n $-quasigroup to a group one. The representation of the $ n $-quasigroup operation proved by the theorem with the help of the automorphism of the group turned out to occur in weaker (and quite natural) assumptions, rather than the associativity and $ (i, j) $-associativity required earlier. Well-known $ (i, j) $-associative $ n $-quasigroups satisfy the condition of weak invertibility.

Keywords: $ n $-quasigroup, $ (i, j) $-associativity, group automorphism, Post–Gluskin–Hoss theorem.

DOI: https://doi.org/10.22405/2226-8383-2018-19-2-304-318

Full text: PDF file (389 kB)
References: PDF file   HTML file

UDC: 512.548.74
Received: 27.04.2018
Accepted:17.08.2018

Citation: F. M. Malyshev, “Weakly invertible $ n $-quasigroups”, Chebyshevskii Sb., 19:2 (2018), 304–318

Citation in format AMSBIB
\Bibitem{Mal18}
\by F.~M.~Malyshev
\paper Weakly invertible $ n $-quasigroups
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 2
\pages 304--318
\mathnet{http://mi.mathnet.ru/cheb656}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-2-304-318}
\elib{https://elibrary.ru/item.asp?id=37112156}


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    This publication is cited in the following articles:
    1. A. V. Cheremushkin, “Partially invertible strongly dependent $n$-ary operations”, Sb. Math., 211:2 (2020), 291–308  mathnet  crossref  crossref  isi  elib
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