Let is a complete lattice. I will call a filter base a nonempty subset of such that . I will call a chain (on ) a linearly ordered subset of .
Now as a part my research of filters I attempt to solve this problem (the problem seems not very difficult and I hope to prove it today or tomorrow, however who knows how difficult it may be):
Definition A subset of a complete lattice is chain-meet-closed iff for every non-empty chain we have .
Conjecture A subset of a complete lattice is chain-meet-closed iff for every filter base we have .