We consider the structure $(\mathbb{ Z},+,0,|_{ p_{ 1}},\dots,|_{ p_{ n}})$, where $x|_{ p}y$ means $v_{ p}(x)\leq v_{ p}(y)$ and $v_p$ is the $p$-adic valuation. We prove that its theory has quantifier elimination in the language $\{ +,-,0,1,(D_{ m})_{ m\geq1},|_{ p_{ 1}},\dots,|_{ p_{ n}}\}$ where $D_m(x)\leftrightarrow \exists y ~ my = x$, and that it has dp-rank $n$. In addition, we prove that a first order structure with universe $\mathbb{ Z}$ which is an expansion of $(\mathbb{ Z},+,0)$ and a reduct of $(\mathbb{ Z},+,0,|_{ p})$ must be interdefinable with one of them. We also give an alternative proof for Conant's analogous result about $(\mathbb{ Z},+,0,<)$.
Categories: math.LO