On Superstable Expansions of Free Abelian Groups

TL;DR

This paper investigates superstable expansions of the integer group, showing that certain expansions with additional predicates remain superstable with finite or infinite Lascar rank, revealing new classes of superstable structures.

Contribution

It proves that $( ext{Z},+,0)$ has no proper superstable finite-rank expansions, but adding specific predicates like powers of a natural number yields superstable structures with infinite rank.

Findings

$( ext{Z},+,0)$ with a predicate for powers of a natural number is superstable of rank $ ext{ extomega}$

Adding factorial elements to $( ext{Z},+,0)$ produces superstable expansions

No proper superstable finite-rank expansions of $( ext{Z},+,0)$ exist

Abstract

We prove that $(\Z,+,0)$ 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 $(\Z,+,0)$ equipped with the set of factorial elements.