# Porton, Victor

## Funcoid corresponding to reloid through lattice Gamma ★★

Author(s): Porton

Conjecture   For every reloid and , :
\item ; \item .

It's proved by me in this online article.

Keywords: funcoid corresponding to reloid

## Restricting a reloid to lattice Gamma before converting it into a funcoid ★★

Author(s): Porton

Conjecture   for every reloid .

## 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

## Coatoms of the lattice of funcoids ★

Author(s): Porton

Problem   Let and be infinite sets. Characterize the set of all coatoms of the lattice of funcoids from to . Particularly, is this set empty? Is a coatomic lattice? coatomistic lattice?

Keywords: atoms; coatoms; funcoids

## Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let be a -separable compact reflexive symmetric funcoid and be a reloid such that
\item ; \item .

Then .

## Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let be a proximity.

A set is connected regarding iff .

Conjecture   The following statements are equivalent for every endofuncoid and a set :
\item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .

Keywords: connected; connectedness; proximity space

## Every monovalued reloid is metamonovalued ★★

Author(s): Porton

Conjecture   Every monovalued reloid is metamonovalued.

Keywords: monovalued

## Every metamonovalued reloid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued reloid is monovalued.

Keywords:

## Every metamonovalued funcoid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued funcoid is monovalued.

The reverse is almost trivial: Every monovalued funcoid is metamonovalued.

Keywords: monovalued

## Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture   For composable reloids and it holds
\item if is a co-complete reloid; \item if is a complete reloid; \item ; \item ; \item .

Keywords: co-completion; completion; reloid