Geodesic
On paratopies of orthogonal systems of ternary quasigroups.~I
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2016), pp. 91-117.

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A paratopy of an orthogonal system Σ={A1,A2,…,An} of n-ary quasigroups, defined on a nonempty set Q, is a mapping θ:Qn↦Qn such that Σθ=Σ, where Σθ={A1θ,A2θ,…,Anθ}. The paratopies of the orthogonal systems, consisting of two binary quasigroups and two binary selectors, have been described by Belousov in [1]. He proved that there exist 9 such systems, admitting at least one non-trivial paratopy and that the existence of paratopies implies (in many cases) the parastrophic-orthogonality of a quasigroup from Σ. A generalization of this result (ternary case) is considered in the present paper. We prove that there exist 153 orthogonal systems, consisting of three ternary quasigroups and three ternary selectors, which admit at least one non-trivial paratopy. The existence of paratopies implies (in many cases) some identities. One of them was considered earlier by T. Evans, who proved that it implies the self-orthogonality of the corresponding ternary quasigroup. The present paper contains the first part of our investigation. We give the necessary and sufficient conditions when a triple θ, consisting of three ternary quasigroup operations or of a ternary selector and two ternary quasigroup operations, defines a paratopy of Σ. 
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P. Syrbu; D. Ceban. On paratopies of orthogonal systems of ternary quasigroups.~I. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2016), pp. 91-117. https://geodesic-test.mathdoc.fr/item/BASM_2016_1_a8/

[1] Belousov V., “Systems of orthogonal operations”, Mat. Sbornik, 77(119):1 (1968), 38–58 (in Russian) | MR | Zbl

[2] Belousov V., “Parastrofic-orthogonal quasigroups”, Quasigroups and Related Systems, 14 (2005), 3–51 | MR

[3] Belousov V., $n$-Ary quasigroups, Shtiintsa, Chisinau, 1972 (in Russian) | MR

[4] Belousov V., Yakubov T., “On orthogonal $n$-ary operations”, Vopr. Kibernetiki, 16, 1975, 3–17 (in Russian)

[5] Evans T., “Latin cubes orthogonal to their transposes – a ternary analogue of Stein quasigroups”, Aequationes Math., 9 (1973), 296–297 | DOI | MR | Zbl

[6] Syrbu P., “On orthogonal and self-orthogonal $n$-ary operations”, Mat. Issled., 66 (1987), 121–129 (in Russian) | MR

[7] Syrbu P., “On $\pi$-quasigroups isotopic to abelian groups”, Bul. Acad. Ştiinţe Repub. Mold., Mat., 2009, no. 3(61), 109–117 | MR | Zbl

[8] Syrbu P., Ceban D., “On $\pi$-quasigroups of type $T_1.$”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2014, no. 2(75), 36–43 | MR | Zbl

[9] Ceban D., Syrbu P., “On the holomorph of $\pi$-quasigroups of type $T_1$”, Proceedings IMCS-50, The 3rd Conference of Math. Society of the Republic of Moldova (Chisinau, August 19–23, 2014), 34–37 | Zbl

[10] Belyavskaya G., “Successively orthogonal systems of $k$-ary operations”, Quasigroups and Related Systems, 22 (2014), 165–178 | MR | Zbl