On the definition of compact funcoids
[I found that my computations below are erroneous, namely $latex \mathrm{Cor} \langle f^{-1}\rangle \mathcal{F} \neq \langle \mathrm{CoCompl} f^{-1}\rangle \mathcal{F}$ in general (the equality holds when $latex \mathcal{F}$ is a set).]
An erroneos theorem (now a conjecture)
In my Algebraic General Topology series was a flaw in the proof of the following theorem. So I re-labeled it as a conjecture. Conjecture A filter $latex \mathcal{A}$ is connected regarding a reloid $latex f$ iff it is connected regarding the funcoid $latex (\mathrm{FCD})f$. Other theorems in my manuscripts are not affected by this error.
Funcoids and Reloids updated – complete reloids
I updated online article “Funcoids and Reloids”. The main feature of this update is new section about complete reloids and completion of reloids (with a bunch of new open problems). Also added some new theorems in the section “Completion of funcoids”.
Funcoids and Reloids – updated
I updated my online article “Funcoids and Reloids”. Now it contains materials which previously were in separate articles: Partially ordered dagger categories; Generalized continuity, which generalizes continuity, proximity continuity, and uniform continuity.
Funcoids and Reloids – completion
I updated the online article “Funcoids and Reloids”. The main feature of this update is introduction of the concepts completion and co-completion of funcoids and some related theorems. The question whether meet (on the lattice of funcoids) of two discrete funcoids is discrete solved positively.
Pointfree Funcoids and Reloids – wiki for writing a math book
I have created a wiki intended to write (collaboratively) a math book about pointfree funcoids and reloids. It is a continuation of my research of Algebraic General Topology (the theory of funcoids and reloids), a generalized point-set topology. The discoverer of funcoids, Victor Porton, does not consider this project “Pointfree Funcoids and Reloids” high priority. […]
“Funcoids and Reloids” update
I updated the online draft of the article “Funcoids and Reloids”. The main feature of this update is that I qualified lattice theoretic operations and direct products of filters with indexes indicating that these are used over the sets of funcoids and reloids. That change should make the article easier to read. But (in a […]
Funcoids and Reloids – two conjectures solved
Updated version of Funcoids and Reloids article contains two counter-examples which constitute solutions of two former open problems. It is proved that: There exist atomic reloids whose composition is non-atomic (and not empty). There exists an atomic reloid which is not monovalued.
Funcoids and Reloids updated to match updated filters theory
I updated the online draft of the article “Funcoids and Reloids” to match updated theory of filters (see also this wiki). The main feature of the new version is using filter objects instead of plain filters. Also in the new version of Funcoids and Reloids removed some theorems which are present in the theory of […]
Funcoids and Reloids – examples of funcoids added
A mathematician has said me that he cannot understand my writings because I introduce new terms without examples before. Because of this I added to my article Funcoids and Reloids (PDF) new subsection Informal introduction into funcoids. Hopefully now it is understandable. Please comment on this message whether you can understand my writings and what […]