General interpolation by polynomial functions of distributive lattices

For a distributive lattice L, we consider the problem of interpolating functions f: D→L defined on a finite set DL n, by means of lattice polynomial functions of L. Two instances of this problem have already been solved. In the case when L is a distributive lattice with least and greatest elements 0 and 1, Goodstein proved that a function f: {0,1} n →L can be interpolated by a lattice polynomial function p: L n →L if and only if f is monotone; in this case, the interpolating polynomial p was shown to be unique. The interpolation problem was also considered in the more general setting where L is a distributive lattice, not necessarily bounded, and where DL n is allowed to range over cuboids with a i,b i L and a i