Let $U$ be some set.
Let $\Gamma$ be the set of all finite unions of cartesian products ($X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$) of sets on $U$.
Obviously, $\Gamma$ is a a distributive lattice.
Let $\mathcal{A}$ and $\mathcal{B}$ be filters on the lattice $\Gamma$. (Note that I consider improper filter to be a filter.)
I denote $\langle K \rangle a = \bigcup \left\{ Y_i \mid i \in \{1,\dots,n\}, X_i \in a \right\}$ where $K = X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$ for every $K \in \Gamma$ and ultrafilter $a$.
Let $\left\{ \langle P \rangle a \mid P \in \mathcal{A} \right\} = \left\{ \langle P \rangle a \mid P \in \mathcal{B} \right\}$ for every ultrafilter $a$ on $U$.
Prove (or give a counter-example) that $\mathcal{A} = \mathcal{B}$.
Consequences of this conjecture (if true) are very important for my research. If you prove it, it is a major step forward in mathematics. Please help.
I proved it with theory of funcoids in http://www.mathematics21.org/binaries/funcoids-are-filters.pdf
I cannot present the proof here, because theory of funcoids is not in common knowledge.