Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Unsorted
 Keywords: complete lattice filter
 Posted by: porton on: July 31st, 2009
 Solved by: Blass, Andreas

Let is a set. A filter (on ) is by definition a non-empty set of subsets of such that . Note that unlike some other authors I do not require . I will denote the lattice of all filters (on ) ordered by set inclusion.

Let is some (fixed) filter. Let . Obviously is a bounded lattice.

I will call complementive such filters that:

1. ;
2. is a complemented element of the lattice .
Conjecture   The set of complementive filters ordered by inclusion is a complete lattice.

To this problem was found a counterexample in a MathOverflow question answer.

The counter example with a rewritten proof is also available in Filters on Posets and Generalizations online article.

Bibliography

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