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 $A\times B$.

Let $L\in\mathscr{P}\Gamma$.

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).

  • $\begingroup$ The conjecture is wrong: Consider $L=\{ \{x\}\times\{y\} \mid x,y\in\mathbb{R} \}$. The question however remains $\endgroup$ – porton Dec 21 '16 at 20:02

It is impossible to determine this property from order-theoretic properties of $L$.

Consider $L_1$ and $L_2$ which are both continuum-size anti-chains (and thus have the same order-theoretic properties), but only $L_1$ not $L_2$ has its union in $\Gamma$.

$L_1=\{ \{x\}\times\{y\} \mid x,y\in\mathbb{R} \}$.

$L_2=\{ \{x\}\times\{y\} \mid x,y\in\mathbb{R}, x\geq y \}$.


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