Arithmetical Congruence Preservation: From Finite to Infinite
نویسندگان
چکیده
Various problems on integers lead to the class of functions defined on a ring of numbers (or a subset of such a rings) METTRE RING AU SINGULIER 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.
منابع مشابه
On the Elasticity of Generalized Arithmetical Congruence Monoids
An arithmetical congruence monoid (or ACM ) is a multiplicative monoid, which consists of an arithmetic sequence and the element 1. As they are traditionally defined, it is required that a ≤ b and a ≡ a (mod b) must hold to ensure closure. It is well known that unique factorization need not occur in ACMs. In this paper, we investigate factorization results when the requirement a ≤ b is dropped....
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