Error corrected
In my draft article Multifuncoids there was a serious error. I defined funcoidal product wrongly. Now a new version of the article (with corrected error) is online.
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 […]
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 […]
On product of an arbitrary family of funcoids
I previously defined product of two funcoids. Now the description of product of two funcoids is integrated into my article about multifuncoids. In this article I now also define product of an arbitrary (possibly infinite) family of multifuncoids. The article is yet a very rough preliminary partial draft. The relations of my definition of product […]
A draft about multifuncoids
I put online a rough preliminary draft about multifuncoids, a generalization of funcoids. It contains a few definitions, and theorems. Probably the most interesting thing in it is what I call graph-composition of multifuncoids. The draft contains several open problems.
Product of two funcoids and product of two funcoids
I’ve put online an article (PDF, a partial draft) where I define product of two morphisms for certain categories. (Such products are pointfree funcoids.) Particularly it is defined product of two funcoids and product of two reloids. It is a more mature version of a draft I put online previously.