A new DP-minimal expansion of the integers

International audienceWe 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 v p 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 $({\Bbb Z}, + ,0, < )$