Logical $k$-valued functions classification based on additive formulae

Function algebra $P_k$ of $k$-valued logic functions and the inclusion lattice ${\cal L}_k$ of closed under superposition classes in $P_k$ are analysed. The classes are described with the use of canonical additive formulae (modulo $k$ sums) of their elements. One summand of each sum is a linear function, the other terms depend on a divisor $d$ of $k$, they determine various families of such classes. For all $k$ and $d$, generating sets and bases of the classes are found, location of each class in ${\cal L}_k$ is determined.