ON SOME IDENTITIES IN TERNARY QUASIGROUPS

Dina CEBAN State University of Moldova

Abstract

Identities of length 5, with two variables in binary quasigroups are called minimal identities. V.Belousov and, independently, F. Bennett showed that, up to the parastrophic equivalence, there are seven minimal identities. The existence of paratopies of orthogonal systems, consisting of two binary quasigroups and the binary selectors, implies three minimal identities (of seven). The existence of paratopies of orthogonal system, consisting of three ternary quasigroups and the ternary selectors, gives 67 identities. In the present article these identities are listed and it is proved that each of 67 identities is equivalent to one of the following four identities: , , , , where is a ternary quasigroup and A necessary condition when a tuple consisting of -ary quasigroups, defined on a set , is a paratopy of the orthogonal system is given. Keywords: minimal identity, -ary quasigroup, paratopy, orthogonal system of quasigroups.