Errors corrected in article “Connectedness of funcoids and reloids”
I have said that there were several errors in my draft article “Connectedness of funcoids and reloids” at Algebraic General Topology site. I have corrected the errors, but now some of what were erroneous theorems downgraded to the status of conjecture.
Errors in my draft article “Connectedness of funcoids and reloids”
In my draft article “Connectedness of funcoids and reloids” at Algebraic General Topology site I found several serious errors. Sorry, I will correct these at some time in the future. (I don’t know how much time will take to find correct proofs of the corrected theorems.) Now I just wrote on the site that it […]
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.