Arithmetical Congruence Preservation: from Finite to Infinite

TL;DR

This paper characterizes congruence preserving functions on rings, linking finite and infinite cases through lifting, and explores their properties on $p$-adic and profinite integers, with applications to recognizable subsets of integers.

Contribution

It provides a unified characterization of congruence preserving functions across finite and infinite rings using lifting techniques.

Findings

Finite case can be lifted to the infinite case.

Characterizations extend to $p$-adic and profinite integers.

Lattices of recognizable subsets are stable under inverse images.

Abstract

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $\pi\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $\pi\circ F=f\circ\pi$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions.