ON PARATOPIES OF ORTHOGONAL SYSTEMS OF TERNARY QUASIGROUPS. I

Abstract

A paratopy of an orthogonal system = {A1,A2, . . . ,An} of n-ary quasigroups, defined on a nonempty set Q, is a mapping : Qn 7→ Qn such that = , where = {A1 ,A2 , . . . ,An }. 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 . A generalization of this result (ternary case) is considered in the present paper. We prove that there exist 153 or- thogonal 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 neces- sary and sufficient conditions when a triple , consisting of three ternary quasigroup operations or of a ternary selector and two ternary quasigroup operations, defines a paratopy of