Geodesic
The Post-Gluskin-Hossz\'u theorem for finite $n$-quasigroups and self-invariant families of permutations
Sbornik. Mathematics, Tome 207 (2016) no. 2, pp. 226-237

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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: associativity, $n$-ary group
Mots-clés : $n$-quasigroup, automorphism, Latin hypercube. 
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     author = {F. M. Malyshev},
     title = {The {Post-Gluskin-Hossz\'u} theorem for finite $n$-quasigroups and self-invariant families of permutations},
     journal = {Sbornik. Mathematics},
     pages = {226--237},
     publisher = {mathdoc},
     volume = {207},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_2_a2/}
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F. M. Malyshev. The Post-Gluskin-Hossz\'u theorem for finite $n$-quasigroups and self-invariant families of permutations. Sbornik. Mathematics, Tome 207 (2016) no. 2, pp. 226-237. http://geodesic.mathdoc.fr/item/SM_2016_207_2_a2/