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A NEW DP-MINIMAL EXPANSION OF THE INTEGERS

Published online by Cambridge University Press:  04 March 2019

ERAN ALOUF
Affiliation:
INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM JERUSALEM91904, ISRAELE-mail: aloufer@gmail.com
CHRISTIAN D’ELBÉE
Affiliation:
INSTITUT CAMILLE JORDAN UNIVERSIT CLAUDE BERNARD LYON 1 69622 VILLEURBANNE, FRANCE and DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS ECOLE NORMALE SUPERIEURE 45 RUE D’ULM, 75005PARIS, FRANCE E-mail: delbee@math.univ-lyon1.fr

Abstract

We consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2019 

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