Category of continuous maps between endofuncoids is cartesian closed
I rough draft article I prove that the category of continuous maps between endofuncoids is cartesian closed. Whether the category of continuous maps between endoreloids is cartesian closed, is yet an open problem.
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]]
Philosophy: God and time machine
I feel that there are certain similarities between God and time machine. Please read and discuss at this tiddler.
A wiki about applying generalized limit to singularities theory
I have created a wiki about development of theory of singularities using generalized limits. Please read my book (where among other I define generalized limits) and then participate in this research wiki. Study singularities in this novel approach and share Nobel Prize with me!
My article has been accepted for publication
Today I’ve received email saying that my article “Funcoids and Reloids: a Generalization of Proximities and Uniformities” has been accepted for publication in European Journal of Pure and Applied Mathematics. See also the preprint of this article.
A new proposition about infimum product
I’ve proved a new simple proposition about infimum product: Theorem Let $latex \pi^X_i$ be metamonovalued morphisms. If $latex S \in \mathscr{P} ( \mathsf{FCD} ( A_0 ; B_0) \times \mathsf{FCD} ( A_1 ; B_1))$ for some sets $latex A_0$, $latex B_0$, $latex A_1$, $latex B_1$ then $latex \bigsqcap \left\{ a \times b \,|\, ( a ; […]
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 […]
Some new simple theorems about funcoids and reloids
I’ve proved the following statements (and put them in my book): Domain of funcoids preserves joins. Image of funcoids preserves joins. Domain of reloids preserves joins. Image of reloids preserves joins. I’ve proved it using Galois connections.
I’ve proved the conjecture about composition of reloids through atomic reloids
From a new version of preprint of my book: Corollary 7.18 If $latex f$ and $latex g$ are composable reloids, then $latex g \circ f = \bigsqcup \left\{ G \circ F \, | \, F \in \mathrm{atoms}\, f, G \in \mathrm{atoms}\, g \right\}$.