# Porton, Victor

## Strict inequalities for products of filters ★

Author(s): Porton

Conjecture   for some filter objects , . Particularly, is this formula true for ?

A weaker conjecture:

Conjecture   for some filter objects , .

Keywords: filter products

## Join of oblique products ★★

Author(s): Porton

Conjecture   for every filter objects , .

Keywords: filter; oblique product; reloidal product

## Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   for every filter objects and and a funcoid ?

Keywords: direct product of filters; outer reloid

## Domain and image of inner reloid ★★

Author(s): Porton

Conjecture   and for every funcoid .

Keywords: domain; funcoids; image; reloids

## Characterization of monovalued reloids with atomic domains ★★

Author(s): Porton

Conjecture   Every monovalued reloid with atomic domain is either
1. an injective reloid;
2. a restriction of a constant function

(or both).

Keywords: injective reloid; monovalued reloid

## Composition of reloids expressed through atomic reloids ★★

Author(s): Porton

Conjecture   If and are composable reloids, then

Keywords: atomic reloids

## Outer reloid of direct product of filters ★★

Author(s): Porton

Question   for every f.o. , ?

Keywords: direct product of filters; outer reloid

## 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