Chain-meet-closed sets on complete lattices

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

Conjecture A subset S of a complete lattice \mathfrak{A} is chain-meet-closed iff for every filter base T\in\mathscr{P}S we have \bigcap T\in S.

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  1. Pingback: Two open problems solved in a new preprint « Victor Porton's Math Blog

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