Consider the structure of the positive real numbers $(0, \infty) $ with its unit $1$, its addition $+$, its multiplication $\times $, and its strict ordering $> $.
Is this structure 
$$( (0, \infty), 1, +, \times, >) $$
o-minimal?



Is the order arithmetic of the positive reals o-minimal?
Another way to see o-minimality is to note that $log$ induces an isomorphism
$$log : ((0,\infty), 1, +, \times, >) \cong (\mathbb R, 0, \oplus, +, >),$$
where $\oplus$ is the binary operation defined by $\oplus(x,y) = \log(e^x + e^y)$. But $\oplus$ is definable in $\mathbb R\_{exp}$. Hence the latter structure is a reduct of $\mathbb R\_{exp}$ and therefore is o-minimal.


