Quasigroup homogeneous spaces and linear representations

Using pseudoinverses of incidence matrices of finite quasigroups in partiti ons induced by left multiplications of subquasigroups, a quasigroup homogen eous space is defined as a set of Markov chain actions indexed by the quasi group. A certain non-unital ring is afforded a linear representation by a q uasigroup homogeneous space. If the quasigroup is a group, the linear repre sentation is a factor in the usual linear representation of the group algeb ra afforded by the group homogeneous space. In the general case, the struct ure of the non-unital ring is analyzed in terms of the permutation action o f the multiplication group of the quasigroup. The linear representation cor estricts to the natural projection of the non-unital ring onto the quotient by its Jacobson radical. (C) 2001 Academic Press.