Arithmetical Congruence Preservation: from Finite to Infinite

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

Integral Difference Ratio Functions on Integers

number theoryInternational audienceTo Jozef, on his 80th birthday, with our gratitude for sharing with us his prophetic vision of Informatique Abstract. Various problems lead to the same class of functions from integers to integers: functions having integral difference ratio, i.e. verifying f (a) − f (b) ≡ 0 (mod (a − b)) for all a > b. In this paper we characterize this class of functions from Z to Z via their a la Newton series expansions on a suitably chosen basis of polynomials (with rational coefficients). We also exhibit an example of such a function which is not polynomial but Bessel like

Continuity of Functional Transducers: A Profinite Study of Rational Functions

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses

On profinite uniform structures defined by varieties of finite monoids

International audienceWe consider uniformities associated with a variety of finite monoids V, but we work with arbitrary monoids and not only with free or free profinite monoids. The aim of this paper is to address two general questions on these uniform structures and a few more specialized ones. A first question is whether these uniformities can be defined by a metric or a pseudometric. The second question is the description of continous and uniformly continuous functions. We first give a characterization of these functions in term of recognizable sets and use it to extend a result of Reutenauer and Schützenberger on continuous functions for the pro-group topology. Next we introduce the notion of hereditary continuity and discuss the behaviour of our three main properties (continuity, uniform continuity, hereditary continuity) under composition, product or exponential. In the last section, we analyse the properties of V-uniform continuity when V is the intersection or the join of a family of varieties and we discuss in some detail the case where V is commutative.Nous considérons les uniformités associées à une variété de monoïdes finis V, mais nous travaillons avec des monoïdes arbitraires et pas seulement avec des monoïdes libres ou profinis libres. On s'intéresse dans cet article à deux questions générales sur ces structures uniformes et à quelques questions plus spécialisées. La première question est de savoir si ces uniformités peuvent être définies par une métrique ou par une pseudométrique. La seconde question est la description des fonctions continues et uniformément continues. Nous donnons d'abord une caractérisation de ces fonctions en termes de parties reconnaissables et nous utilisons ce résultat pour étendre un résultat de Reutenauer et Schützenberger sur les fonctions continues pour la topologie pro-groupe. Nous introduisons ensuite la notion de continuité héréditaire et nous discutons la préservation de nos trois principales propriétés (continuité, uniforme continuité, continuité héréditaire) par composition, produit et exponentiation. Dans la dernière partie, nous analysons les propriétés de V-uniforme continuité lorsque V est l'intersection ou l'union d'une famille de variétés et nous discutons en détail du cas où V est commutative