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Chain-meet-closed sets (Solved)

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Unsorted
Keywords: chain
complete lattice
filter bases
filters
linear order
total order
Recomm. for undergrads: no
Posted by: porton
on: December 12th, 2009
Solved by: Joel David Hamkins

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) $.

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 $.

The answer is yes. A proof is present in this online article.

Bibliography

*Victor Porton. Chain-meet-closed sets on complete lattices


* indicates original appearance(s) of problem.
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