Generalizing a theorem about filters on a boolean lattice

Question

Let $\mathfrak{A}$ be a bounded distributive lattice with binary meet and join $\sqcap$ and $\sqcup$.

I will denote $\partial F = \{ X\in\mathfrak{A} \mid \forall Y\in F: X\sqcap Y\ne \bot \}$ where $F$ is a filter on $\mathfrak{A}$ and $\bot$ is the least element of the lattice.

I will say that the set of filters is star-separable iff $A\ne B$ implies $\partial A \ne \partial B$ for every filters $A$ and $B$.

I my draft book it is proved that the set of filters on a boolean lattice is star-separable.

Can this result be strengthened? Particularly, is it true for all bounded distributive lattices?

Answer 1

This is not true for all bounded distributive lattices. Consider the distributive lattice $\bot \to a \to \top$ and the filters $\uparrow{a}$ and $\{\top\}$.