Category: Open problems

Oblique products, related theorems and conjectures

I updated Funcoids and Reloids article. Now it contains a section on oblique products. It now contains also the following conjectures: Conjecture $latex \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B}$ for some f.o. $latex \mathcal{A}$, $latex \mathcal{B}$. Conjecture $latex \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}$ for some f.o. $latex \mathcal{A}$, […]

Uniformization of funcoids

I’ve put a sketch draft of future research “Uniformization of funcoids” on my Algebraic General Topology page. “Uniformization” is meant to be a generalization of oblique products of filters. This future research may be published as a part of my Funcoids and Reloids article, or (what is more likely) as a separate article.

Oblique product of filters

Funcoids and reloids are my research in the field of general topology. Let $latex \mathcal{A}$ and $latex \mathcal{B}$ are filters. Earlier I introduced three kinds of products of filters: funcoidal product: $latex \mathcal{A}\times^{\mathsf{FCD}}\mathcal{B}$; reloidal product: $latex \mathcal{A}\times^{\mathsf{RLD}}\mathcal{B}$; second product: $latex \mathcal{A}\times^{\mathsf{RLD}}_F\mathcal{B}$; The last two products are reloids while the first is a funcoid. Funcoidal and […]

A surprisingly hard problem

I am now trying to prove or disprove this innocently looking but somehow surprisingly hard conjecture: Conjecture If $latex S$ is a generalized filter base then $latex \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S$ for every funcoid $latex f$.

Two propositions and a conjecture

I added to Funcoids and Reloids article the following two new propositions and a conjecture: Proposition $latex (\mathsf{FCD}) (f\cap^{\mathsf{RLD}} ( \mathcal{A}\times^{\mathsf{RLD}} \mathcal{B})) = (\mathsf{FCD}) f \cap^{\mathsf{FCD}} (\mathcal{A}\times^{\mathsf{FCD}} \mathcal{B})$ for every reloid $latex f$ and filter objects $latex \mathcal{A}$ and $latex \mathcal{B}$. Proposition $latex ( \mathsf{RLD})_{\mathrm{in}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{in}} f) […]

Changes in “Funcoids and Reloids”

I added one new proposition and two open problems to my online article “Funcoids and Reloids”: Conjecture $latex \left\langle f \right\rangle \mathcal{X} = \bigcap^{\mathfrak{F}} \left\{ \left\langle F \right\rangle \mathcal{X} | F \in \mathrm{up}f \right\}$ for every funcoid $latex f$ and f.o. $latex \mathcal{X}$. Proposition $latex \mathrm{dom}( \mathsf{\mathrm{FCD}}) f =\mathrm{dom}f$ and $latex \mathrm{im}(\mathsf{\mathrm{FCD}}) f =\mathrm{im}f$ for […]

Question: Complete classification of ultrafilters?

Are there a known complete classification of filters (or at least ultrafilters)? By complete classification I mean a characterization of every filter by a family of cardinal numbers such that two filters are isomorphic if and only if they have the same characterization. For definition of isomorphic filters see my article “Filters on Posets and […]

Two new conjectures in “Funcoids and Reloids” article

Though my Funcoids and Reloids article was declared as a preprint candidate, I made a substantial addendum to it: Added definitions of injective, surjective, and bijective morphisms. Added a conjecture about expressing composition of reloids through atomic reloids. Added a conjecture characterizing monovalued reloids with atomic domains.