# Porton, Victor

## 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

## 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

## Monovalued reloid restricted to atomic filter ★★

Author(s): Porton

Conjecture   A monovalued reloid restricted to an atomic filter is atomic or empty.

Weaker conjecture:

Conjecture   A (monovalued) function restricted to an atomic filter is atomic or empty.

Keywords: monovalued reloid

## Atomic reloids are monovalued ★★

Author(s): Porton

Conjecture   Atomic reloids are monovalued.

Keywords: atomic reloid; monovalued reloid; reloid

## Composition of atomic reloids ★★

Author(s): Porton

Conjecture   Composition of two atomic reloids is atomic or empty.

Keywords: atomic reloid; reloid

## S(S(f)) = S(f) for reloids ★★

Author(s): Porton

Question   for every endo-reloid ?

Keywords: reloid

## Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

Conjecture   For any funcoid and reloid having the same source and destination

Keywords: funcoid; inward reloid; outward reloid; reloid

## Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

Conjecture   if is a set of reloids from a set to a set .

Keywords: funcoid; infinite distributivity; reloid

## Inward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

Conjecture   for any convex reloid .

Keywords: convex reloid; funcoid; functor; inward reloid; reloid

## Outward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

Conjecture   for any convex reloid .

Keywords: convex reloid; funcoid; functor; outward reloid; reloid