Interpolation by polynomial functions of distributive lattices : a generalization of a theorem of R. L. Goodstein

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein’s theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, resp.: Given a function f : {0, 1}n → L, there exists a lattice polynomial function p : Ln → L such that p|{0,1}n = f if and only if f is monotone; in this case, the interpolating polynomial p is unique. We extend Goodstein’s theorem to a wider class of partial functions f : D → L over a distributive lattice L, not necessarily bounded, and where D ⊆ Ln is allowed to range over n-dimensional rectangular boxes D = {a1, b1} × · · · × {an, bn} with ai, bi ∈ L and ai < bi, and determine the class of such partial functions which can be interpolated by lattice polynomial functions. In this wider setting, interpolating polynomials are not necessarily unique; we provide explicit descriptions of all possible lattice polynomial functions which interpolate these partial functions, when such an interpolation is available.