Quasigroup Homogeneous Spaces and Linear Representations

Using pseudoinverses of incidence matrices of finite quasigroups in partitions induced by left multiplications of subquasigroups, a quasigroup homogeneous space is defined as a set of Markov chain actions indexed by the quasigroup. A certain non-unital ring is afforded a linear representation by a quasigroup homogeneous space. If the quasigroup is a group, the linear representation is a factor in the usual linear representation of the group algebra afforded by the group homogeneous space. In the general case, the structure of the non-unital ring is analyzed in terms of the permutation action of the multiplication group of the quasigroup. The linear representation corestricts to the natural projection of the non-unital ring onto the quotient by its Jacobson radical.