On Dualization over Distributive Lattices

Given a partially order set (poset) $P$, and pair of families ideals $\mathcal{I}$ filters $\mathcal{F}$ in $P$ such that each $(I,F)\in \mathcal{I}\times\mathcal{F}$ has non-empty intersection, the dualization problem over is to check whether there an ideal $X$ which intersects every member does not contain any $\mathcal{I}$. Equivalently, for distributive lattice $L=L(P)$, given by poset its joint-irreducibles, two antichains $\mathcal{A},\mathcal{B}\subseteq L$ no $a\in\mathcal{A}$ dominated $b\in\mathcal{B}$, $\mathcal{A}$ $\mathcal{B}$ cover (by domination) entire lattice. We show can be solved quasi-polynomial time sizes $\mathcal{B}$, thus answering open question Babin Kuznetsov (2017). As application, we minimal infrequent closed sets attributes rational database, with respect implication base maximum premise size one, enumerated incremental time.