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Preprint Number 1262
1262. Eran Alouf and Christian d'Elbée A new minimal expansion of the integers E-mail: Submission date: 22 July 2017 Abstract: We consider the structure (Z,+,0,|_{p_{1}},...,|_{p_{n}}), where x|_{p}y means v_{p}(x) ≤ v_{p}(y). We prove that its theory has QE in the language {+,-,0,1,(D_{m})_{m ≥ 1},|_{p_{1}},...,|_{p_{n}}), and that it has dp-rank n. In addition, we prove that a first order structure with universe Z which is an expansion of (Z,+,0) and a reduct of (Z,+,0,|_{p}) must be interdefinable with one of them. We also give an alternative proof for Conant's analogous result about (Z,+,0,<). Mathematics Subject Classification: Keywords and phrases: |

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