# 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 .

Keywords: chain; complete lattice; filter bases; filters; linear order; total order

## 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:

- ;
- is a complemented element of the lattice .

**Conjecture**The set of complementive filters ordered by inclusion is a complete lattice.

Keywords: complete lattice; filter