The starting point of this paper is the characterization of the compound-Poisson, i.e. the infinitely divisible lattice distributions (class C_1) and the subset of compound-geometric lattice distributions (class C_0) by the nonnegativity of recursively defined quantities (section 1). In section 2 we generalize these recursion relations by introducing sequences c_n (\alpha) for \alpha \in [0,1] and thus we obtain classes of distributions C_\alpha, which we wish to be increasing with \alpha. This can be achieved by an appropriate choice of c_n(\alpha) (section 3). For the classes C_\alpha we obtain other properties, which generalize known properties of the class C_1. Another subdivision of this class is given in section 4, where we consider the compound-negative-binomial lattice distributions.