On Superstable Expansions of Free Abelian Groups

Abstract

We prove that $(\mathbb{Z},+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $\omega$. Additionally, our methods yield other superstable expansions such as $(\mathbb{Z},+,0)$ equipped with the set of factorial elements.