Generalizations of Khovanskii's theorems on growth of sumsets in abelian semigroups

We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $\chi$ is a coloring of the lattice points in the orthant such that the color $\chi(a+b)$ depends only on the colors $\chi(a)$ and $\chi(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\u\i{} on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\u\i's theorem. Another result of Khovanski\u\i{} states that the size of the image of a finite set after $n$ applications of mappings from a finite family of mutually commuting mappings is for large $n$ a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.