TITLE: Quantifier elimination for $\mathbb Z$ as a group?
QUESTION [6 upvotes]: Question 1: What definable sets should one add to the language to obtain quantifier elimination for the theory of $(\mathbb Z, +)$, i.e. the integers as a group (short of simply Morleyizing)?
An $\omega$-saturated model is given by $\hat{\mathbb Z} \oplus {\mathbb Q}^{\oplus \omega}$. This can be seen by showing that if a constant for 1 is added to the language, then after defining negation and $n | x := \exists y (ny = x)$ for each $n \in \mathbb N$, one can perform a back-and-forth argument for the structure $(\hat{\mathbb Z} \oplus {\mathbb Q}^{\oplus \omega}, 0, 1, + , -, (n|)_{n \in \mathbb N})$ (where $1$ is interpreted as $1 \in \hat{\mathbb Z}$), so that this structure eliminates quantifiers; it is easy to see that it realizes all quantifier-free types, so it is $\omega$-saturated. Taking a reduct, $\hat{\mathbb Z} \oplus {\mathbb Q}^{\oplus \omega}$ is $\omega$-saturated in the group language.
However, $1$ is not definable in $(\mathbb Z, +)$, so this doesn't tell us how to obtain quantifier elimination in a definitional extension of the group language. A related puzzle is
Question 2: What is a model of the theory of $(\mathbb Z,+)$ which omits the type of $1$?

REPLY [3 votes]: In his answer, Alex Kruckman points out the excellent reference: Eklof and Fischer, The elementary theory of abelian groups. Let me just spell out what can be gleaned from this. They recover Szmielew's result that any abelian group $A$ is characterized up to elementary equivalence by the following four families of elementary invariants:

The torsionfree invariants $Tf(p; A) := \lim_{n \to \infty} \operatorname{dim}_{\mathbb F_p}(\frac{p^n A}{p^{n+1} A})$;

The order invariants $D(p; A) := \lim_{n \to \infty} \operatorname{dim}_{\mathbb F_p}((p^n A)[p])$;

The global order invariant $Exp(A)$, which is $0$ if $A[n] = A$ for some $n$ and $\infty$ otherwise;

The Ulm invariants $U(p,n;A) := \operatorname{dim}_{\mathbb F_p}(\frac{(p^n A)[p]}{(p^{n+1} A)[p]})$.


Here $p$ ranges over primes, $n \in \mathbb N$, and $B[n] = \{b \in B \mid nb = 0\}$ is the $n$-torsion of an abelian group $B$.
This simplifies nicely for us because in a torsionfree group, only the torsionfree invariants are nontrivial, and moreover the limit in the definition of the torsionfree invariant stabilizes at $n=0$. Thus the theory of $(\mathbb Z,+)$ is the theory of a torsionfree abelian group $A$ such that $A / p \cong \mathbb Z/p$ for each prime $p$.
It follows, then, along with the quantifier elimination results, that if $A$ is a model of the theory of $\mathbb Z$ and $B \subseteq A$ is any pure subgroup (i.e. a subgroup with the same $n$-divisibility relation as $A$ has for each $n$) which is not $p$-divisible for any $p$, then $B$ is an elementary subgroup of $A$ (the reasoning is that if $B/p = 0$, then $B$ is $p$-divisible, while if $B/p$ is bigger than $A/p$ then purity is contradicted). In particular, Alex's example of a subgroup of $\hat {\mathbb{Z}}$ omitting the type of $1$ is indeed a model of the theory of $\mathbb Z$.