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:

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Full text arXiv 1707.07203: pdf, ps.