Quadratical quasigroups and Mendelsohn designs

Drápal, Aleš; Griggs, Terry S. and Kozlik, Andrew R. (2022). Quadratical quasigroups and Mendelsohn designs. International Journal of Algebra and Computation (Early Access).

DOI: https://doi.org/10.1142/s0218196722500308

Abstract

Let the product of points A and B be the vertex C of the right isosceles triangle for which AB is the base, and ABC is oriented anticlockwise. This yields a quasigroup that satisfies laws (xu) (vy) = (xv)(uy), (xy)(yx) = y and xx = x. Such quasigroups are called quadratical. Quasigroups that satisfy only the latter two laws are equivalent to perfect Mendelsohn designs of length four (PMD(v,4)). This paper examines various algebraic identities induced by PMD(v,4), classifies finite quadratical quasigroups, and shows how the square structure of quadratical quasigroups is associated with toroidal grids.

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