Let $latex \mathfrak{A}$ is a complete lattice. I will call a filter base a nonempty subset $latex T$ of $latex \mathfrak{A}$ such that $latex \forall a,b\in T\exists c\in T: (c\le a\wedge c\le b)$. I will call a chain (on $latex \mathfrak{A}$) a linearly ordered subset of $latex \mathfrak{A}$.
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 $latex S$ of a complete lattice $latex \mathfrak{A}$ is chain-meet-closed iff for every non-empty chain $latex T\in\mathscr{P}S$ we have $latex \bigcap T\in S$.
Conjecture A subset $latex S$ of a complete lattice $latex \mathfrak{A}$ is chain-meet-closed iff for every filter base $latex T\in\mathscr{P}S$ we have $latex \bigcap T\in S$.
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