Does there exist a semi-ring $(R,+,\cdot)$ (like a ring, but there must be no additive inverses and the $0$ is multiplicatively absorbing by axiom) with isomorphic additive and multiplicative structures? This means there should be a monoid-isomorphism
$$\varphi:(R,+)\longleftrightarrow(R,\cdot).$$
First thing I noticed is that there must be an absorbing element for the addition, lets call it $\omega=\varphi^{-1}(0)$. There follow many more such "strange" elements, e.g. $\varphi^{-1}(\omega)$, etc. Because addition and multiplication are so similar, the multiplication is automatically commutative and has an identity element $1$. But this is essentially as far as I came.
If the answer to above question is "Yes", then it would be very interesting to see if also $\Bbb N$ can be extended to such a semi-ring.
Note that any natural number $n$ can be represented as a sum of two natural numbers in exactly $\lceil (n+1)/2\rceil$ ways. So for any $n\in\Bbb N$ there are at most two natural numbers that can be represented in exactly $n$ different ways as sum of two other natural numbers. But there are infinitely many numbers that can be written as product of two natural numbers in exactly $n$ ways. This means that this extension of $\Bbb N$ must contain many additional elements.