ON PROFINITE UNIFORM STRUCTURES DEFINED BY VARIETIES OF FINITE MONOIDS

Abstract

We 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 analyze 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.