Latin Squares over Quasigroups

We propose a construction that allows generating large families of Latin squares, i.e., Cayley tables of finite quasigroups. This construction generalizes proper families of functions over Abelian groups introduced by Nosov and Pankratiev. We also show that all quasigroups generated by the original construction contain at least one subquasigroup, while the generalized construction generates quasigroups free of subquasigroups.