# Porton, Victor

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

Author(s): Porton

**Conjecture**for every reloid .

Keywords: funcoid corresponding to reloid; funcoids; reloids

## 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?

## 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 .

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

## 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

## A construction of direct product in the category of continuous maps between endo-funcoids ★★★

Author(s): Porton

Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.

Remind from my book that morphisms of this category are defined by the formula (here and below by abuse of notation I equate functions with corresponding principal funcoids).

Let are endofuncoids,

We define

(here and are cartesian projections).

**Conjecture**The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in

**Set**).

Keywords: categorical product; direct product