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

Abstract

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, respectively: given a function f : {0, 1}ⁿ → L , there exists a lattice polynomial function ${p: L^{n} \rightarrow L}$ such that p| ⁿ_{0,1} =  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 ${f : D \rightarrow L}$ over a distributive lattice L, not necessarily bounded, and where ${{D}\, {\subseteq}\, {L}^{n}}$ is allowed to range over n-dimensional rectangular boxes ${D = \{{a_{1}, b_{1}}\} {\times}. . . {\times} \{{a_{n}, b_{n}\}}}$ with ${a_{i}, b_{i} \in L}$ and ${a_{i} < b_{i}}$ , and we determine the class of such partial functions that 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 that interpolate these partial functions, when such an interpolation is available.