TITLE: Structures strictly between $\{\mathbb{R},+,<\}$ and $\{\mathbb{R},+,\cdot,<\}$.
QUESTION [6 upvotes]: Let $\mathcal{R}_0=\{\mathbb{R},+,<\}$ be the order divisible abelian group of reals and $\mathcal{R}=\{\mathbb{R},+,\cdot,<\}$ be the real closed field of reals. Both $\mathcal{R}_0$ and $\mathcal{R}$ are o-minimal structures. 
Let $f:\mathbb{R}\setminus\{0\}\rightarrow \mathbb{R}$ be the map given by $f(x)=1/x$. Consider $\mathcal{R}_f=\{\mathbb{R},+,f,<\}$, the structure generated after expanding $\mathcal{R}_0$ by adding function $f$ as a definable set. 

Is $\mathcal{R}_f = \mathcal{R}$, i.e. is the field product in $\mathbb{R}$ definable in $\mathbb{R}_f$?

The question arises from the more general doubt of whether there exists an o-minimal group containing a definable homeomorphism between a bounded and an unbounded set that doesn't end up defining also a field structure.

REPLY [5 votes]: Multiplication is definable in $\mathcal R_f$.
Step 0. All the additive group operations ($+, -, 0$) are definable.
The operation $+$ is given in the language, while 
$0$ is the unique identity
for $+$ and $-x$ is the unique additive inverse of $x$.
Step 1. The singleton $\{1\}$ is definable.
$1$ is the unique element $x$ satisfying
the formula $(0<x)\wedge (f(x)=x)$.
Step 2. The squaring function $y = x^2$ is definable.
$x^2=y$ holds exactly when $x=\pm 1$ and $y=1$, or else
$x\neq \pm 1$ and 
$$
y = \frac{1}{\frac{1}{x-1} - \frac{1}{x+1}} + \frac{1}{\frac{1}{x-1} - \frac{1}{x+1}} + 1,
$$
which is expressible using only the additive structure, $1$, and $f(x) = \frac{1}{x}$.
Step 3.  The product function $z = xy$ is definable.
$xy = z$ holds exactly when $(x+y)^2 = x^2 + y^2 + z+z$,
which is expressible using only the additive structure
and squaring.