On paratopies of orthogonal systems of ternary quasigroups. I

A paratopy of an orthogonal system $\Sigma=\{A_1,A_2,\dots,A_n\}$ of $n$-ary quasigroups, defined on a nonempty set $Q$, is a mapping $\theta\colon Q^n\mapsto Q^n$ such that $\Sigma\theta=\Sigma$, where $\Sigma\theta=\{A_1\theta, A_2\theta,\dots,A_n\theta\}$. 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 $\Sigma$. 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 $\theta$, consisting of three ternary quasigroup operations or of a ternary selector and two ternary quasigroup operations, defines a paratopy of $\Sigma$.