On dp-minimal expansions of the integers

Eran Alouf

On dp-minimal expansions of the integers

Logic 2024-12-25 v2

Authors: Eran Alouf

Abstract

We show that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ that defines an infinite subset of $\mathbb{N}$ , then $\mathcal{Z}$ is interdefinable with $\left(\mathbb{Z},+,0,1, < \right)$ . As a corollary, we show the same for dp-minimal expansions of $\left(\mathbb{Z},+,0,1\right)$ which do not eliminate $\exists^{\infty}$ .

Keywords

dp-minimal structures expansions of integers interdefinability model theory exists-infinity

Cite

@article{arxiv.2001.11480,
  title  = {On dp-minimal expansions of the integers},
  author = {Eran Alouf},
  journal= {arXiv preprint arXiv:2001.11480},
  year   = {2024}
}

Comments

21 pages, final version incorporating referee comments

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