= Prime-sign automorphism of the multiplicative integers
{title2=$s_p(n)=(-1)^{v_p(n)}n$}
For a fixed prime $p$, put $s_p(0)=0$ and $s_p(n)=(-1)^{v_p(n)}n$ for $n\ne0$, 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.
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