$k$-homogeneous Latin Squares, their transversals and condition of pseudo-orthogonality

We describe a method for transforming any system of $t$ mutually orthogonal Latin squares into a system of $t$ mutually pseudo-orthogonal Latin squares. We consider the $k$-homogeneous Latin Squares, i.e. Latin Squares of order $kn$ with elements from ${0,\dots,kn-1}$ such that after reducing modulo $n$ we obtain $(kn\times kn)$-matrix consisting of $k^2$ identical Latin Squares of order $n$. Some characteristics of transversals of $k$-homogeneous Latin Squares are described. Sufficient condition of pseudo-orthogonality is presented.