New easy theorem and its consequence
I’ve added to my book a new easy to prove theorem and its corollary: Theorem If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f$ is a filter. Corollary If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f = \mathrm{up} (\mathsf{RLD})_{\mathrm{out}} f$. If $latex f$ is a (co-)complete reloid then $latex \mathrm{up}\, […]
A new conjecture
While writing my book I forgot to settle the following conjecture: Conjecture $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex f$ and $latex g$. Note that the similar statement about reloids is quite obvious.
A new (but easy to prove) theorem in my research book: Theorem Let $latex \mu$ and $latex \nu$ be endomorphisms of some partially ordered dagger precategory and $latex f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu)$ be a monovalued, entirely defined morphism. Then $latex f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).$
$T_4$-funcoids
I have added to my free ebook a definition of $latex T_4$-funcoids (generalizing $latex T_4$ topologies). A funcoid $latex f$ is $latex T_4$ iff $latex f \circ f^{- 1} \circ f \circ f^{- 1} \sqsubseteq f \circ f^{- 1}$. This can also be generalized for pointfree funcoids.
Reexamined: Normal quasi-uniformity elegantly defined
Earlier I’ve conceived an algebraic formula to characterize whether a quasi-uniform space is normal (where normality is defined in Taras Banakh sense, not in the sense of underlying topology being normal). That my formula was erroneous. Today, I have proved another formula for this (hopefully now correct): Theorem An endoreloid $latex f$ is normal iff […]
New easy theorem and a couple of conjectures
I added to my free ebook a new (easy to prove) theorem and a couple new conjectures: Theorem For every reloid $latex f$: $latex \mathsf{Compl} (\mathsf{FCD}) f = (\mathsf{FCD}) \mathsf{Compl} f$; $latex \mathsf{CoCompl} (\mathsf{FCD}) f = (\mathsf{FCD}) \mathsf{CoCompl} f$. Conjecture $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{in}} g=(\mathsf{RLD})_{\mathsf{in}}\mathsf{Compl} g$; $latex \mathsf{Compl}(\mathsf{RLD})_{\mathsf{out}} g=(\mathsf{RLD})_{\mathsf{out}}\mathsf{Compl} g$;.
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.