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 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 $({\Bbb Z}, +,0, < )$.