# Porton, Victor

## Several ways to apply a (multivalued) multiargument function to a family of filters ★★★

Author(s): Porton

**Problem**Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?

1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .

2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .

3. .

Keywords: funcoid; function; multifuncoid; staroid

## Which outer reloids are equal to inner ones ★★

Author(s): Porton

Warning: This formulation is vague (not exact).

**Question**Characterize the set . In other words, simplify this formula.

The problem seems rather difficult.

Keywords:

## A diagram about funcoids and reloids ★★

Author(s): Porton

Define for posets with order :

- ;
- .

Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).

Then we have the following diagram:

What is at the node "other" in the diagram is unknown.

**Conjecture**"Other" is .

**Question**What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?

Keywords: Galois connections

## Outward reloid of composition vs composition of outward reloids ★★

Author(s): Porton

**Conjecture**For every composable funcoids and

Keywords: outward reloid

## A funcoid related to directed topological spaces ★★

Author(s): Porton

**Conjecture**Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.

**Proposition**It is easy to prove that is the infinitely small right neighborhood filter of point .

If proved true, the conjecture then can be generalized to a wider class of posets.

Keywords:

## Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

**Conjecture**for principal funcoid and a set of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

## Entourages of a composition of funcoids ★★

Author(s): Porton

**Conjecture**for every composable funcoids and .

Keywords: composition of funcoids; funcoids

## What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let be an indexed family of sets.

*Products* are for .

*Hyperfuncoids* are filters on the lattice of all finite unions of products.

**Problem**Is a bijection from hyperfuncoids to:

- \item prestaroids on ; \item staroids on ; \item completary staroids on ?

If yes, is defining the inverse bijection? If not, characterize the image of the function defined on .

Consider also the variant of this problem with the set replaced with the set of complements of elements of the set .

Keywords: hyperfuncoids; multidimensional

## Another conjecture about reloids and funcoids ★★

Author(s): Porton

**Definition**for reloid .

**Conjecture**for every funcoid .

Note: it is known that (see below mentioned online article).

Keywords: