A new dp-minimal expansion of the integers

Abstract

We consider the structure $(\mathbb{Z},+,0,{|}_{p_1},\dots ,{|}_{p_n})$, where $x{|}_{p}y$ means $v_p(x)\le v_p(y)$ and $v_p$ is the $p$-adic valuation. We prove that its theory has quantifier elimination in the language $\{+,-,0,1,(D_m{)}_{m\ge 1},{|}_{p_1},\dots ,{|}_{p_n}\}$ where $D_m(x)\leftrightarrow \exists ymy=x$, and that it has dp-rank $n$. In addition, we prove that a first order structure with universe $\mathbb{Z}$ which is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,0,{|}_p)$ must be interdefinable with one of them. We also give an alternative proof for Conant's analogous result about $(\mathbb{Z},+,0,<)$.

24 pages