# Porton, Victor

## 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

## Distributivity of a lattice of funcoids is not provable without axiom of choice ★

Author(s): Porton

**Conjecture**Distributivity of the lattice of funcoids (for arbitrary sets and ) is not provable in ZF (without axiom of choice).

A similar conjecture:

**Conjecture**for arbitrary filters and on a powerset cannot be proved in ZF (without axiom of choice).

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

## Values of a multifuncoid on atoms ★★

Author(s): Porton

**Conjecture**for every multifuncoid of the form whose elements are atomic posets.

Keywords:

## A conjecture about direct product of funcoids ★★

Author(s): Porton

**Conjecture**Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

## Graph product of multifuncoids ★★

Author(s): Porton

**Conjecture**Let is a family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .

Keywords: graph-product; multifuncoid

## Atomicity of the poset of multifuncoids ★★

Author(s): Porton

**Conjecture**The poset of multifuncoids of the form is for every sets and :

- \item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Atomicity of the poset of completary multifuncoids ★★

Author(s): Porton

**Conjecture**The poset of completary multifuncoids of the form is for every sets and :

- \item atomic; \item atomistic.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords: multifuncoid

## Upgrading a completary multifuncoid ★★

Author(s): Porton

Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .

**Conjecture**If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

Keywords:

## Funcoidal products inside an inward reloid ★★

Author(s): Porton

**Conjecture**(solved) If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

A stronger conjecture:

**Conjecture**If then for every funcoid and , .

Keywords: inward reloid

## Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

**Conjecture**for any composable funcoids and .

Keywords: distributive; distributivity; funcoid; functor; inward reloid; reloid