Numeration systems on a regular language: arithmetic operations, recognizability and formal power series

Abstract

Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L⊂Σ ∗ . For these systems, we obtain a characterization of recognizable sets of integers in terms of N -rational formal series. After a study of the polynomial regular languages, we show that, if the complexity of L is Θ(n l) (resp. if L is the complement of a polynomial language), then multiplication by λ∈ N preserves recognizability only if λ=β l+1 (resp. if λ≠(#Σ) β ) for some β∈ N . Finally, we obtain sufficient conditions for the notions of recognizability for abstract systems and some positional number systems to be equivalent.