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This article is cited in 5 scientific papers (total in 5 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.
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”, Sb. Math., 207:2 (2016), 226–237
Linking options:
https://www.mathnet.ru/eng/sm8447https://doi.org/10.1070/SM8447 https://www.mathnet.ru/eng/sm/v207/i2/p81
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Abstract page: | 521 | Russian version PDF: | 146 | English version PDF: | 21 | References: | 122 | First page: | 73 |
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