# 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:

- ;
- 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

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

Author(s): Porton

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