Normal quasi-uniformity elegantly defined
Error in the proof of below theorem!! http://www.math.portonvictor.org/binaries/addons.pdf My earlier attempt to describe normal reloids (where the normality is taken from “Each regular paratopological group is completely regular” article, not from the customary normality of the induced topological space) failed. After a little more thought, I have however proved (see addons.pdf) a cute algebraic description […]
About “Each regular paratopological group is completely regular” article
In this blog post I consider my attempt to rewrite the article “Each regular paratopological group is completely regular” by Taras Banakh, Alex Ravsky in a more abstract way using my theory of reloids and funcoids. The following is a general comment about reloids and funcoids as defined in my book. If you don’t understand […]
Interior funcoid definition
Having a funcoid $latex f$ I defined what I call interior funcoid $latex f^\circ$. See this PDF file for a few equivalent definitions. Interior funcoid is a generalization of interior operator for a topological space. Interior funcoid is kinda dual funcoid. Note that $latex f^{\circ\circ} = f$.
A new easy theorem about pointfree funcoids
I have added the following easy to prove theorem to my general topology research book: Theorem If $latex \mathfrak{A}$ and $latex \mathfrak{B}$ are bounded posets, then $latex \mathsf{pFCD}(\mathfrak{A}; \mathfrak{B})$ is bounded.
An important conjecture about funcoids. Version 2
This conjecture appeared to be false. Now I propose an alternative conjecture: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is nonempty. […]
An important conjecture about funcoids
Just a few minutes ago I’ve formulated a new important conjecture about funcoids: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is […]
Join of transitive reloids (a conjecture in uniformity theory)
Conjecture Join of a set $latex S$ on the lattice of transitive reloids is the join (on the lattice of reloids) of all compositions of finite sequences of elements of $latex S$. It was expired by theorem 2.2 in “Hans Weber. On lattices of uniformities”. There is a similar conjecture for funcoids (instead of reloids). […]
Preservation of properties of funcoids and reloids by their relationships
I have added a new section “Properties preserved by relationships” to my math research book. This section considers (in the form of theorems and conjectures) whether properties (reflexivity, symmetry, transitivity) of funcoids and reloids are preserved an reflected by their relationships (functions $latex (\mathsf{FCD})$, $latex (\mathsf{RLD})_{\mathrm{in}}$, $latex (\mathsf{RLD})_{\mathrm{out}}$ which map between funcoids and reloids).
New version of my math research book
I’ve released a new version of my free math ebook. The main feature of this new release is chapter “Alternative representations of binary relations” where I essentially claim that the following are the same: binary relations pointfree funcoids between powersets Galois connections between powersets antitone Galois connections between powersets This theorem is presented with a […]
A new math abstraction, categories of sides
I introduce a new math abstraction, categories of sides, in order to generalize two theorems into one. Category of sides $latex \Upsilon$ is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables): $latex (a \sqcup […]