A New Dp-Minimal Expansion of the Integers

Abstract

We show that if Z is a dp-minimal expansion of (Z, +, 0, 1) that defines an infinite subset of N, then Z is interdefinable with (Z, +, 0, 1, <). As a corollary, we show the same for dp-minimal expansions of (Z, +, 0, 1) which do not eliminate ∃ ∞ .

There Are No Intermediate Structures Between the Group of Integers and Presburger Arithmetic

“…We claim that (x, y) ∈ B if and only if (1)x ∈ (A) and y ∈ mZ and In particular, since (A) is definable in (Z, +, 0) by assumption, this shows that B is definable in (Z, +, 0). To verify that the above data defines B, fix (x, y) ∈ B.…”

“…By Theorem 1.1 however, there are no such structures other than (Z, +, 0) and (Z, +, <, 0), and so a positive answer to Question 1.2(b) would require a genuinely different dp-minimal expansion of (Z, +, 0). In fact, such expansions have recently been discovered by Alouf and D'Elbée [1].…”

Generalizing a theorem of Bès and Choffrut

“…Conant's proof relies on a detailed analysis of (Z, <, +)-definable subsets of Z n . Alouf and d'Elbée [1] give a quicker proof. They use deep model-theoretic machinery to reduce to the unary case and thereby avoid geometric complexity.…”

“…(2) If I is a bounded interval then the structure induced on I by R is o-minimal. It is clear that (2) implies (1), the other direction follows by compactness of closed bounded intervals. It follows from independent work of Miller [16] or Weispfenning [27] that (R, <, +, Z) is locally o-minimal, another example of a locally ominimal non-o-minimal structure is (R, <, +, sin), see [21].…”

A P-adic structure which does not interpret an infinite field but whose Shelah completion does

“…(1) Suppose that O is trace definable in M and P is trace definable in O. Then P is trace definable in…”