Complete Sets of Orthogonal Self-Orthogonal Latin Squares

We show how to produce algebraically a complete orthogonal set of Latin squares from a left quasifield and how to generate algebraically a maximal set of self-orthogonal Latin squares from a left nearfield. For a left Veblen-Wedderburn system, we establish the algebraic relationships between the standard projective plane construction of a complete set of Latin squares, our projective plane construction, and our algebraic nearfield generation. Via a projective plane construction, we establish the equivalence of a complete set of self-orthogonal Latin squares and a restricted ((0), ∞)-Desarguesian plane.