This article is cited 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.

UDC: 512.548.74

MSC: Primary 20N15; Secondary 20N05

Received: 11.11.2014 and 20.05.2015

DOI: 10.4213/sm8447

 English version: Sbornik: Mathematics, 2016, 207:2, 226–237

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