# complete lattice

## Chain-meet-closed sets ★★

Author(s): Porton

Let is a complete lattice. I will call a filter base a nonempty subset of such that .

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 .

## Do filters complementive to a given filter form a complete lattice? ★★

Author(s): Porton

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.

Keywords: complete lattice; filter