A topological approach to transductions

This paper is a contribution to the mathematical foundations of the theory of automata. We give a topological characterization of the transductions τ from a monoid M into a monoid N, such that if R is a recognizable subset of N,τ-1(R) is a recognizable subset of M. We impose two conditions on the monoids, which are fullfilled in all cases of practical interest: the monoids must be residually finite and, for every positive integer n, must have only finitely many congruences of index n. Our solution proceeds in two steps. First we show that such a monoid, equipped with the so-called Hall distance, is a metric space whose completion is compact. Next we prove that τ can be lifted to a map τ^ from M into the set of compact subsets of the completion of N. This latter set, equipped with the Hausdorff metric, is again a compact monoid. Finally, our main result states that τ-1 preserves recognizable sets if and only if τ^ is continuous.