Arithmetical congruence preservation: From finite to infinite

Abstract

Various problems on integers lead to the class of functions defined on a ring of numbers (or a subset of such a ring) and verifying a−b divides f(a)−f(b) for all a, b.We say that such functions are “congruence preserving”. In previous works, we characterized these classes of functions for the cases ℕ → ℤ, ℤ → ℤ and ℤ/nℤ → ℤ/mℤ in terms of sums series of rational polynomials (taking only integral values) and the function giving the least common multiple of 1, 2,..., k. In this paper we relate the finite and infinite cases via a notion of “lifting”: if π: X → Y is a surjective morphism and f is a function Y → Y a lifting of f is a function F: X → X such that π ◦ F = f ◦ π. We prove that the finite case ℤ/nℤ → ℤ/nℤ can be so lifted to the infinite cases ℕ → ℕ and ℤ → ℤ. We also use such liftings to extend the characterization to the rings of p-adic and profinite integers, using Mahler representation of continuous functions on these rings.