Let $A$ and $B$ two finite sets of natural numbers and by $\mathcal{P}(X)$ we denote the set of all subsets of a given set $X$. Find the number of functions $f: \mathcal{P}(A) \to \mathcal{P}(B)$, for which $f(X) \cap f(Y) = f(X \cap Y)$.
I think that I might associate each subset of $A$ with its product of elements, while considering, without loss of generality, that each of the elements are prime numbers.
This equivalence might be seen as the number of injections (from the identity consideration) between a set of $n$ prime numbers and the set of all possible products of these prime numbers, which are at most $2^n - 1$ elements. So, the number would be: $$N = \prod_{i = 1}^n (2^n - i)$$
Is this a correct reasoning?