There is given family of sets, $\mathcal{F}\subseteq \mathcal{P}(\{1,2,...,m\})$.
$\mathcal{P}(A)$ is a set of all the subsets of $A$.
Consider now an arbitrary set $S\in \mathcal{P}(\{1,2,...,m\})$.
I would like to find an equivalent test to determine whether $$S\in \{X\cap Y: X\in \mathcal{F}, Y\in \mathcal{F}\}$$.
How to do this?
I look for a simple algebraic tool, that could be useful. I have tried to associate vectors to each set or to create a logical matrix, but this gave me no result.
Regards.