Prime-sign automorphism of the multiplicative integers ($s_p(n)=(-1{)}^{v_p(n)}n$)

For a fixed prime $p$, put $s_p(0)=0$ and $s_p(n)=(-1{)}^{v_p(n)}n$ for $n\ne 0$, where $v_p(n)$ is its P-adic valuation. Additivity of the valuation on products proves $s_p(mn)=s_p(m)s_p(n)$. Changing sign does not change the valuation, so $s_p^2$ is the identity. Thus this is a structure automorphism of $(\mathbb{Z},\cdot )$. At $p=2$ it fixes one and sends two to minus two, so it does not preserve the addition graph or the nonnegative integers. Automorphism invariance of parameter-free definability therefore excludes both from definability in this multiplicative language.