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