On Self-Orthogonal n-ary Quasigroups [Articol]

Abstract

We consider self-orthogonal finite n-ary quasigroups and give some estimations of their spectrum in the present work. A method of construction of self-orthogonal n-quasigroups, using self-orthogonal quasigroups of lower arity, is given. In particular, it is shown that there exist: 1) self-orthogonal 2k-ary quasigroups of every order q > 3; q ≠ 6, where k ≥ 1; 2) self-orthogonal pk quasigroups of prime order p, for every p ≥ 3 and every k ≥ 1; 3) self-orthogonal 2n-quasigroups of order q, for every q > 3; q ≠ 6 and n + 1 ≢ 0 (mod q).