Parastrophic-orthogonal Quasigroups

Annotated translation of Parastrophic-orthogonal quasigroups, Acad. Nauk Moldav. SSR, Inst. Mat.s Vychisl. Tsentrom, Kishinev, 1983, prepared by A.D.Keedwell and P.Syrbu based on the original Russian and on an earlier English translation supplied to the rst author by Belousov himself. The notion of orthogonality plays an important role in the theory of Latin squares, and consequently also in the theory of quasigroups, because every nite quasigroup has a Latin square as its Cayley table and, conversely, every Latin square is the multiplication table of a certain quasigroup. The concept of orthogonality can be described very easily in algebraic language. Two quasigroups Q(A), Q(B) (i.e. quasigroups with operations A and B de ned on the same set Q) are orthogonal if the system of equations A(x, y) = a, B(x, y) = b has a unique solution for every pair of elements a, b ∈ Q. There is signi cant interest in the investigation of quasigroups orthogonal to their parastrophes (for the de nitions see below). However, in the past, the questions mainly considered have been some combinatorial ones which have arisen in connection with these investigations. We mention, for example, the Phelps papers (for example [6]) which are devoted to the study of the spectrum of α-orthogonal quasigroups, i.e. quasigroups A which are orthogonal to their parastrophe αA. Special cases of such quasigroups were considered earlier in connection with other problems having purely algebraic character. For example, Stein 2000 Mathematics Subject Classi cation: 20N15, 20N05