Some model theory of $\operatorname{Th}(\mathbb{N},\cdot)$

TL;DR

This paper characterizes the definably stable sets in the theory of natural numbers with multiplication, revealing the unique role of squarefree numbers and analyzing the theory's elimination properties of imaginaries.

Contribution

It provides a complete description of definably stable sets in Skolem arithmetic and clarifies the theory's elimination of imaginaries properties.

Findings

The set of squarefree numbers is the only non-trivial definably stable set.

The theory has weak elimination of imaginaries.

The theory does not have elimination of finite imaginaries.

Abstract

'Skolem arithmetic' is the complete theory $T$ of the multiplicative monoid $(\mathbb{N},\cdot)$. We give a full characterization of the $\varnothing$-definable stably embedded sets of $T$, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that $T$ has weak elimination of imaginaries but not elimination of finite imaginaries.