Some families of closed classes in $P_k$ defined by additive formulas

We analyse closed classes in $k$-valued logics containing all linear functions modulo $k$. The classes are determined by divisors $d$ of a number $k$ and canonical formulas for functions. We construct the lattice of all such classes for $k=p^2$, where $p$ is a prime, and construct fragments of the lattice for other composite $k$.