It is not hard to see that 3-SAT
reduces to the problem of deciding whether all coatoms in a certain partition
lattice are contained in the union of a collection of certain principal filters.
Therefore, the latter problem, which we will call the
*covered coatoms problem* (CCP), is NP-complete.
In this post we simply define CCP. Later we check that 3-SAT reduces to CCP,
and then develop some ideas about constructing a feasible algorithm to solve CCP.