On the spectrum of Stein quasigroups

Abstract

In this paper we investigate the spectrum of a variety of quasigroups satisfying the 2-variable identity x(xy) = yx, called Stein quasigroups. Stein quasigroups are known to be self-orthogonal and have been given a considerable amount of attention because of this property. It is known that there are no Stein quasigroups of order 2, 3, 6, 7, 8, 10, 12, 14. The object of this paper is to show that for all but 36 values of n ≥ 15 there exists a Stein quasigroup of order n. In particular, the spectrum of Stein quasigroups contains all n ≥ 191.