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 $(\mathbb{Z},+,0,<)$.

arXiv comment24 pages

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Cited by in corpus (8)

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• Interpolative Fusions I
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• Generalizing a theorem of Bès and Choffrut
• Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang
• A P-adic structure which does not interpret an infinite field but whose Shelah completion does
• Dp-minimal profinite groups and valuations on the integers