I’ve solved two yesterday problems, one yet remains unsolved
I have solved the first two of these three open problems I proposed, but have no clue how to solve the third. (Actually, I’ve solved only a special case of the second problem, but that’s OK, this special case is enough for all practical needs.) The solutions are in this article. I asked about the […]
Three new conjectures
See here (especially this draft article) for definition of cross-composition product and quasi-cartesian functions. Conjecture 1 Cross-composition product (for small indexed families of relations) is a quasi-cartesian function (with injective aggregation) from the quasi-cartesian situation $latex {\mathfrak{S}_0}&fg=000000$ of binary relations to the quasi-cartesian situation $latex {\mathfrak{S}_1}&fg=000000$ of pointfree funcoids over posets with least elements. Conjecture […]
Abrupt categories induced by categories with star-morphisms
In this blog post I introduced the notion of category with star-morphisms, a generalization of categories which have aroused in my research. Each star category gives rise to a category (abrupt category, see a remark below why I call it “abrupt”), as described below. Below for simplicity I assume that the set $latex {M}&fg=000000$ and […]
Categories with star-morphisms, a generalization of categories
In my research aroused a new kind of structures which I call categories with star-morphisms. In this blog post I define categories with star-morphisms. For sample usages of star categories see this draft article. Definition 1 A pre-category with star-morphisms consists of a pre-category $latex {C}&fg=000000$ (the base pre-category); a set $latex {M}&fg=000000$ (star-morphisms); a […]
Multidimensional Funcoids – draft available
The status of the article “Multidimensional Funcoids” is raised from “very rough partial draft” to “rough partial draft”. It means that now you probably can understand this my writing. See my research in general topology.
A difficulty on the way of my research
The following conjecture seems trivial but I have a hard hour trying to prove it. I suspect I have a big difficulty on the course of my research. Conjecture $latex \prod^{\mathsf{FCD}} a \not\asymp\prod^{\mathsf{FCD}} b \Leftrightarrow \forall i\in n : a_i \not\asymp b_i$ for every $latex n$-indexed (where $latex n$ is an arbitrary index set) families […]
Candidate formulas for product of reloids
First, we can define product of reloids as a trivial generalization of the alternative definition of product of uniform spaces. There are no trivial simplification of this relatively inelegant definition, it is not algebraic as I would want. I (without any evidence or intuition) propose two open questions one of which may be true despite […]
The end of Algebraic General Topology?
It was my long time dream since the first course of the university to do math analysis in algebraic fashion instead of the notorious epsilon-delta notation overloaded with quantifiers. This dream has been accomplished. I discovered a new math theory in which I call Algebraic General Topology in the field of General Topology. It all […]
“Upgrading multifuncoid” conjecture proved
I have proved the conjecture “Upgrading a multifuncoid is multifuncoid”. The proof is currently presented in this online draft article.
The article about ordinal numbers and product
I’ve sent this article to an open access peer reviewed journal: A New Kind of Product of Ordinal Number of Relations having Ordinal Numbers of Arguments (the article is available online).