Let $U$ be some set.
Let $\Gamma$ be the set of all finite unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y_X\in\mathscr{P}U$ for every $X\in S$ (that is $Y$ is a function from $S$ to $\mathscr{P}U$).
Obviously, $\Gamma$ is a a distributive lattice.
Let $\Gamma^{-1}$ be the result of reversing x and y in every element of $\Gamma$.
What is the sublattice of the lattice of all binary relations on $U$ generated by the sublattices $\Gamma$ and $\Gamma^{-1}$?