ON SOME IDENTITIES IN TERNARY QUASIGROUPS

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 paratopiesof orthogonal systems,consistingof 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 equivalentto one of the following four identities: α A ( β A, γ A,δ A)= E1, α A ( β A, γ A,E1) = E2, : α A (β A, E1, E2) = , γ A (β A, E1, E3), α A ( β A, E1, E2 ) = γ A (β A, E1 E2) , where A is a ternary quasigroup and ÷α,β,γ,δ∈〖S4〗_ necessary condition when a tuple θ = ( A1 ,A, 2, …,A n ) consisting of n-ary quasigroups, defined on a set Q2 Место для формулы., is a paratopy of the orthogonal system ∑{ A1, A2, …, An, E1, E2, …, En} is given