On distribution of some classes of monotone $k$-valued functions on the lattice of closed classes

In this paper we consider one of the families of classes of functions of multivalued logics closed with respect to superposition, namely the family of monotone classes. We study the question how these classes are situated on the lattice of all closed classes of functions in the case where they are not precomplete. We describe the family of monotone classes over which infinite chains of closed classes are distributed, and prove that in the case of monotone classes of functions which preserve a partially ordered set with one minimal or one maximal element the minimal multi-valued logic with such a class in an infinite depth is $P_5$.