We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

This paper is devoted to understand groups definable in Presburger Arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded abelian group definable in a model of (Z, +, <) Presburger Arithmetic is definably isomorphic to (Z, +)^n mod out by a lattice.

Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger. In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming that the induced theories on H and are definable complete and (...) o‐minimal respectively, we prove that any definable set of G is ‐definable. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset [math] and every interpretable continuous function [math] (where [math] denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as a (...) consequence of their work. The first one shows that every interpretable subset of [math] is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P-minimal field is polynomially bounded. The second one characterizes those P-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues. (shrink)

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.