Is there a set theoretic construction of the natural numbers or integers such that the product of two numbers is their Cartesian product?

Question

Is there a set theoretic construction of the natural numbers or integers such that the product of two numbers is their Cartesian product? What I mean is, e.g., if $25 = A$ and $2 = B$ then $50 = A\times B = \{(a,b)\mid a\in A \land b\in B\}$.

Answer 1

If this property holds, then at most one number is represented by the empty set. Let $n,k$ be two numbers which are not empty sets.

We have that $n\times k=k\times n$. Therefore $(a,b)$ appears in both and so $a\in n$ implies that $a\in k$ and similarly $b\in k$ implies that $b\in n$. So $n=k$.

This means that all the numbers which are non-empty sets are equal, which is impossible.