A conjecture related with subatomic product
With subatomic products first mentioned here and described in this article are related the following conjecture (or being precise three conjectures): Conjecture For every funcoid $latex f: \prod A\rightarrow\prod B$ (where $latex A$ and $latex B$ are indexed families of sets) there exists a funcoid $latex \Pr^{\left( A \right)}_k f$ defined by the formula $latex […]
A new theorem about subatimic product
I recently discovered what I call subatomic product of funcoids. Today I proved a simple theorem about subatomic product: Theorem $latex \prod^{\left( A \right)}_{i \in n} \left( g_i \circ f_i \right) = \prod^{\left( A \right)} g \circ \prod^{\left( A \right)} f$ for indexed (by an index set $latex n$) families $latex f$ and $latex g$ […]
Subatomic products – a new kind of product of funcoids
I’ve discovered a new kind of product of funcoids, which I call subatomic product. Definition Let $latex f : A_0 \rightarrow A_1$ and $latex g : B_0 \rightarrow B_1$ are funcoids. Then $latex f \times^{\left( A \right)} g$ (subatomic product) is a funcoid $latex A_0 \times B_0 \rightarrow A_1 \times B_1$ such that for every […]
A conjecture about direct product of funcoids
I am attempting to define direct products in the category cont(mepfFcd) (the category of monovalued, entirely defined continuous pointfree funcoids), see this draft article for a definition of this category. A direct product of objects may possibly be defined as the cross-composition product (see this article). A candidate for product of morphisms $latex f_1:\mathfrak{A}\rightarrow\mathfrak{B}$ and […]
“Categories related with funcoids”, a new draft
I started to write a new article Categories related with funcoids. It is now a very preliminary partial draft.
Is every isomorphisms of the category of funcoids a discrete funcoid?
The following is an important question related with categories related with funcoids: Question Is every isomorphisms of the category of funcoids a discrete funcoid?
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 […]