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