On Interpretations of Presburger Arithmetic in Büchi Arithmetics

Abstract

Büchi arithmetics BA_n, \(n \geqslant 2\), are extensions of Presburger arithmetic with an unary functional symbol \({{V}_{n}}(x)\) denoting Presburger arithmetic the largest power of n that divides x. Definability of a set in BA_n is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We consider the interpretations of in the standard model of BA_n and show that each such interpretation has an internal model isomorphic to the standard one. This answers a question by A. Visser on the interpretations of certain weak arithmetical theories in themselves.