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

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

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UDC: 512.548.74
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
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