# filters

## Inner reloid through the lattice Gamma ★★

Author(s): Porton

Conjecture   for every funcoid .

Counter-example: for the funcoid is proved in this online article.

Keywords: filters; funcoids; inner reloid; reloids

## 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 .

## Co-separability of filter objects ★★

Author(s): Porton

Conjecture   Let and are filters on a set and . Then

See here for some equivalent reformulations of this problem.

This problem (in fact, a little more general version of a problem equivalent to this problem) was solved by the problem author. See here for the solution.

Maybe this problem should be moved to "second-tier" because its solution is simple.

Keywords: filters

## Pseudodifference of filter objects ★★

Author(s): Porton

Let is a set. A filter (on ) is a non-empty set of subsets of such that . Note that unlike some other authors I do not require .

I will call the set of filter objects the set of filters ordered reverse to set theoretic inclusion of filters, with principal filters equated to the corresponding sets. See here for the formal definition of filter objects. I will denote the filter corresponding to a filter object . I will denote the set of filter objects (on ) as .

I will denote the set of atomic lattice elements under a given lattice element . If is a filter object, then is essentially the set of ultrafilters over .

Problem   Which of the following expressions are pairwise equal for all for each set ? (If some are not equal, provide counter-examples.)
\item ;

\item ;

\item ;

\item .

Keywords: filters; pseudodifference