2011/10/03 by Miguel Couceiro, Couceiro, Miguel, Tamás Waldhauser +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Mathematical Identities #Coding theory and cryptography #FOS: Mathematics #Mathematical Analysis and Transform Methods #Rings and Algebras (math.RA) #math.RA #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1110.0321
12 pages
arxiv created 2011/10/03 · openalex publication_date 2011/10/03 · arxiv updated 2011/10/04 · openalex created_date 2023/02/28 · openalex updated_date 2026/07/28
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 an n-ary partial function f over L, defined on all 0-1 tuples, f can be extended to a lattice polynomial function p over L 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 n-ary partial functions f over a distributive lattice L, not necessarily bounded, where the domain of f is a cuboid of the form D=a1,b1x...xan,bn with 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.