Right distributive quasigroups on algebraic varieties

Abstract

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes $$\mathfrak{D}$$ generating algebraic Fischer groups (in the sense of [6]) such that the mappingx ↦x −1 ax, fora e $$\mathfrak{D}$$ , is an automorphism of $$\mathfrak{D}$$ (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product. We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.