Definable Types in Presburger Arithmetic

We consider the first order theory of (Z,+, <), also known as Presburger arithmetic. We prove a characterization of definable types in terms of prime models over realizations, which has a similar flavor to the Marker-Steinhorn Theorem of o-minimality. We also prove that a type over a model is definable if and only if it has a unique coheir to any elementary extension, which is a characterization of definable types that Presburger arithmetic shares with stable theories and o-minimal theories. 1. Presburger Arithmetic Let L = {+,−, 0, 1, <} and let TPr = Th(Z,+,−, 0, 1, <). Note that all integers in Z are definable from + and <. We specify the following definable relations: (1) For n ∈ Z, a unary predicate Pn defining the elements divisible by n, i.e., TPr |= ∀x(Pn(x)↔ ∃y ny = x). (2) For n ∈ Z, a binary relation ≡n defining congruence mod n, i.e. TPr |= ∀x∀y(x ≡n y ↔ Pn(x− y)). Note that nx is shorthand for x+ . . .+ x (n times). We will also use n|x to mean Pn(x). Theorem 1.1. [2] TPr has quantifier elimination in the language L ∪ {Pn : n ∈ Z}. See [2, Section 3.1] for more details on TPr, including a full axiomatization in L ∪ {Pn : n ∈ Z}. An important property of TPr, which is shown in [1] and follows easily from quantifier elimination, is that it is quasi-o-minimal, i.e., for any M |= TPr, any definable subset of M (in one dimension) is a Boolean combination of ∅-definable sets and intervals with endpoints in M. For the rest of this section, let M |= TPr. Definition 1.2. Let a, b ∈M with a < b. For n ∈ Z, set |b− a| = n if M |= ∃!n−1x a < x < b. If there is no such n, set |b− a| =∞. Set |b− a| = 0 for a = b; and let |a− b| = |b− a|. Date: March 26, 2013. 1