A new dp-minimal expansion of the integers

TL;DR

This paper introduces a new dp-minimal expansion of the integers involving p-adic valuations, proving quantifier elimination, dp-rank, and structural interdefinability results, advancing understanding of dp-minimal structures.

Contribution

It establishes a novel dp-minimal expansion of the integers with quantifier elimination and dp-rank n, and characterizes structures between additive and valuation expansions.

Findings

The theory has quantifier elimination in the specified language.

The dp-rank of the structure is n.

Any expansion of $(Z,+,0)$ that reduces to the valuation structure is interdefinable with one of them.

Abstract

We consider the structure $(\mathbb{Z},+,0,|_{p_{1}},\dots,|_{p_{n}})$, where $x|_{p}y$ means $v_{p}(x)\leq v_{p}(y)$ and $v_p$ is the $p$-adic valuation. We prove that its theory has quantifier elimination in the language $\{+,-,0,1,(D_{m})_{m\geq1},|_{p_{1}},\dots,|_{p_{n}}\}$ where $D_m(x)\leftrightarrow \exists y ~ my = x$, 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 $(\mathbb{Z},+,0,<)$.