“Funcoidal” reloids, a new research idea
Just today I’ve got the idea of the below conjecture: Definition I call funcoidal such reloid $latex \nu$ that $latex \mathcal{X} \times^{\mathsf{RLD}} \mathcal{Y} \not\asymp \nu \Rightarrow \\ \exists \mathcal{X}’ \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{X})} \setminus \{ 0 \}, \mathcal{Y}’ \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{Y})} \setminus \{ 0 \} : ( \mathcal{X}’ \sqsubseteq \mathcal{X} \wedge \mathcal{Y}’ \sqsubseteq \mathcal{Y} […]
Cauchy spaces and Cauchy continuity expressed through reloids
In the march of development of my theory, I have expressed many kinds of spaces (topological, proximity, uniform spaces, etc.) through funcoids and reloids subsuming their properties (such as continuity) for my algebraic operations. Now I have expressed Cauchy spaces (and some more general kinds of spaces) through reloids. And yes, Cauchy continuity appears as […]
New theorem about funcoids and reloids
I’ve added to the preprint of my book a new theorem (currently numbered theorem 8.30). The theorem states: Theorem $latex g \circ ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) \circ f = \langle ( \mathsf{FCD}) f^{- 1} \rangle \mathcal{A} \times^{\mathsf{RLD}} \langle ( \mathsf{FCD}) g \rangle \mathcal{B}$ for every reloids $latex f$, $latex g$ and filters $latex \mathcal{A} \in […]
Cauchy filters, a generalization for reloids
I have proved the theorem from Wikipedia that every Cauchy filter is contained in a maximal Cauchy filter (in fact I’ve proved a more general statement). I don’t know the standard proof (and don’t know where to find it), so I’ve devised a proof myself. I hope (thanks to use the notation reloids in my […]
An error in my book
In the draft of my book there was an error. I’ve corrected it today. Wrong: $latex \forall a, b \in \mathfrak{A}: ( \mathrm{atoms}\, a \sqsubset \mathrm{atoms}\, b \Rightarrow a \subset b)$. Right: $latex \forall a, b \in \mathfrak{A}: ( a \sqsubset b \Rightarrow \mathrm{atoms}\, a \subset \mathrm{atoms}\, b)$. There are the same error in my […]
A filter which cannot be partitioned into ultrafilters
I’ve proved: There exists a filter which cannot be (both weakly and strongly) partitioned into ultrafilters. It is an easy consequence of a lemma proved by Niels Diepeveen (also Karl Kronenfeld has helped me to elaborate the proof). See the preprint of my book.
A negative result on a conjecture
Due my research about singularities the problem formulated in this blog post was solved negatively with help of Alex Ravsky who has found a counter-example. The conjecture was: $latex \mathrm{GR}(\Delta \times^{\mathsf{FCD}} \Delta)$ is closed under finite intersections. The counter-example follows: $latex f=\{(x,y)\in\mathbb R^2:|x|\le |y| \vee y=0\}$, $latex g=\{(x,y)\in\mathbb R^2:|x|\ge |y| \vee x=0\}$. It is easy […]
Changes to my article about products in certain categories
There are two changes in Products in dagger categories with complete ordered Mor-sets draft article: 1. I’ve removed the section on relation of subatomic product with categorical product saying that for funcoids they are the same. No, they are not the same. My claim that they are the same was false. 2. Added section “Special […]
A suggested way to solve an open problem
On the task formulated in this blog post: An attempt to prove that $latex \mathrm{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ is closed under finite intersections (see http://portonmath.tiddlyspace.com/#[[Singularities%20funcoids%3A%20some%20special%20cases]]) http://portonmath.tiddlyspace.com/#[[Singularities%20funcoids%3A%20special%20cases%20proof%20attempts]]
Some new minor results
I’ve proved: $latex \bigsqcap \langle \mathcal{A} \times^{\mathsf{RLD}} \rangle T = \mathcal{A} \times^{\mathsf{RLD}} \bigsqcap T$ if $latex \mathcal{A}$ is a filter and $latex T$ is a set of filters with common base. $latex \bigsqcup \left\{ \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B} \hspace{1em} | \hspace{1em} \mathcal{B} \in T \right\} \neq \mathcal{A} \times^{\mathsf{RLD}} \bigsqcup T$ for some filter $latex T$ and […]