Let $A$, $B$ be (infinite) sets. Denote $\Gamma$ the set of finite unions of binary cartesian products of subsets of $A$ and $B$. That is $P = X_0\times Y_0\cup\dots X_n\times Y_n$ where $X_i\subseteq A$, $Y_i\subseteq B$ for every element $P\in\Gamma$.
Notice (easy to prove) that $\Gamma$ is a boolean sublattice of the lattice of all subsets of $A\times B$.
Question: Determine (if possible) if the union $L$ is a member of the lattice $\Gamma$, using only order-theoretic properties of $L$.
Conjecture: The union $L$ is a member of the lattice $\Gamma$ if and only if $L$ can be partitioned into a finite number of chains (totally ordered subsets).