$k$-splitted and $k$-homogeneous Latin squares and their transversals

We consider the $k$-splitted Latin squares, i.e. Latin squares of order $kn$ with elements from $\left\{ {0, \ldots ,kn - 1} \right\}$ such that after reducing modulo $n$ we obtain $\left( {kn \times kn} \right)$-matrix consisting of $k^2$ Latin squares of order $n$. If these $k^2$ Latin squares of order $n$ are identical, the original Latin square of order $kn$ is called $k$-homogeneous. The precise number of all $k$-homogeneous and lower bound for the number of all $k$-splitted Latin squares are found. Some characteristics of transversals for $k$-splitted Latin squares are described.