Geodesic
The Post-Gluskin-Hosszú theorem for finite $n$-quasigroups and self-invariant families of permutations
Sbornik. Mathematics, Tome 207 (2016) no. 2, pp. 226-237 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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F. M. Malyshev. The Post-Gluskin-Hosszú 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/

[1] A. G. Kurosh, Obschaya algebra, Lektsii 1969–1970 uchebnogo goda, Nauka, M., 1974, 159 pp. | MR | Zbl

[2] V. D. Belousov, $n$-arnye kvazigruppy, Shtiintsa, Kishinev, 1972, 227 pp. | MR | Zbl

[3] A. M. Galmak, $n$-arnye gruppy, v. 1, GGU im. F. Skoriny, Gomel, 2003, 196 pp.

[4] E. L. Post, “Polyadic groups”, Trans. Amer. Math. Soc., 48:2 (1940), 208–350 | DOI | MR | Zbl

[5] L. M. Gluskin, “Pozitsionnye operativy”, Matem. sb., 68(110):3 (1965), 444–472 | MR | Zbl

[6] M. Hosszú, “On the explicit form of $n$-group operations”, Publ. Math. Debrecen, 10 (1963), 88–92 | MR | Zbl

[7] W. Dörnte, “Untersushungen über einen verallgeweinerten Gruppenbegriff”, Math. Z., 29:1 (1928), 1–19 | DOI | MR | Zbl

[8] A. M. Galmak, G. N. Vorobev, “O teoreme Posta–Gluskina–Khossu”, PFMT, 2013, no. 1(14), 55–60 | Zbl

[9] F. N. Sokhatskii, “Ob assotsiativnosti mnogomestnykh operatsii”, Diskret. matem., 4:1 (1992), 66–84 | MR | Zbl

[10] E. I. Sokolov, “O teoreme Gluskina–Khossu dlya $n$-grupp Dernte”, Matem. issledovaniya, 39, Shtiintsa, Kishinev, 1976, 187–189 | MR | Zbl

[11] A. M. Galmak, N. A. Schuchkin, “Porozhdayuschie mnozhestva $n$-arnykh grupp”, Chebyshevskii sb., 15:1 (2014), 89–109 | MR

[12] M. Hosszú, “Algebrai rendszereken értelmezett függvényegyenletek. I”, Magyar Tud. Akad. Mat. Fiz. Oszt. Kozl., 12 (1962), 303–315 | MR | Zbl

[13] V. V. Borisenko, “O transversalyakh raspavshikhsya latinskikh kvadratov chetnogo poryadka”, Matem. vopr. kriptogr., 5:1 (2014), 5–25