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 $latex g \circ f = \bigsqcup \left\{ G \circ F \, | \, F \in \mathrm{atoms}\, f, G \in \mathrm{atoms}\, g \right\}$.
My gibberish with partial proofs
I’ve put online my gibberish with partial proofs and proof attempts of my open problems. You can see the PDF file with this gibberish. Please write me (either by email or by blog comments) if you solve anything of this.
Co-equalizers and arbitrary co-limits exist in categories Fcd and Rld
I’ve uploaded a new version of my article “Equalizers and co-Equalizers in Certain Categories” (a very rough draft). In it is proved (among other) that arbitrary equalizers and co-equalizers of categories Fcd and Rld (continuous maps between endofuncoids and endoreloids) exist. I’ve proved earlier that in these categories there exist products and co-products. So now […]
Equalizers in certain categories
In this my rough draft article I construct equalizers for certain categories (such as the category of continuous maps between endofuncoids). Products and co-products were already proved to exist in my categories, so these categories are complete. In the above mentioned article I also claim co-equalizers (however have not yet proved that they are really […]
A few new conjectures
Conjecture 1. The categories Fcd and Rld are complete and co-complete (actually 4 conjectures). I have not yet spend much time trying to solve this conjecture, it may be probably easy. Conjecture 2. The categories Fcd and Rld are cartesian closed (actually two conjectures).
New definitions of products and coproducts in certain categories
I have updated this article. It now contains a definition of product and coproduct for arbitrary morphisms of a dagger category every of Hom-sets of which is a complete lattice. Under certain conditions these products and coproducts are categorical (co)products for a certain category (“category of continuous morphisms”) having endomorphisms of the aforementioned category as […]
Product and co-product of endoreloids is now defined
Using this recently proved theorem I have defined product and co-product of endo-reloids. It is expressed by elegant algebraic formulas.
Monovalued reloids are metamonovalued
I’ve proved today the theorem: Theorem Monovalued reloids are metamonovalued. In other words: Theorem $latex \left( \bigsqcap G \right) \circ f = \bigsqcap \left\{ g \circ f \,|\, g \in G \right\}$ if $latex f$ is a monovalued reloid and $latex G$ is a set of reloids (with matching sources and destination). The proof uses […]
New chapter in my book
I have added new chapter: 9 “On distributivity of composition with a principal reloid” into my research monograph preprint. (Read the book) The chapter is centered over a single theorem that composition with a principal reloid is distributive over join of reloids. It also defines an embedding from the set of reloids to the set […]