# 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 .

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

## 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