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 […]
Embedding reloids into funcoids
I have codified my idea how to embed reloids into funcoids in this draft article. Next I am going to attempt to solve some of my conjectures using this embedding. I will announce in this blog how solving the open problems goes. For the moment, I have also noticed a new problem: Conjecture $latex \rho […]
A draft proof of distributivity of composition with a principal reloid over join of reloids
Recently I’ve announced that I have an elegant proof idea of this conjecture, but have a trouble to fill in details of the proof: Statement Composition with a principal reloid is distributive over join of reloids. Now I have almost complete draft proof of the above statement (Well, it is yet to be checked for […]
Decomposition of composition and a partial proof of a conjecture
Composition of binary relations can be decomposed into two operations: $latex \otimes$ and $latex \mathrm{dom}$: $latex g \otimes f = \left\{ ( ( x ; z) ; y) \, | \, x f y \wedge y g z \right\}$. Composition of binary relations is decomposed as: $latex g \circ f = \mathrm{dom} (g\otimes f)$. I […]
A brilliant idea about funcoids and reloids
In my book I introduced concepts of funcoids and reloids. To every funcoid $latex f$ it corresponds a reloid $latex (\mathsf{RLD})_{\mathrm{in}}f$. This allows to represent a funcoid as a reloid. Today I had the thought that it would be good also to represent a reloid as a funcoid. After not so long thinking I realized: […]
Direct products in a category of funcoids
I’ve released a draft article about categorical products and coproducts of endo-funcoids, as well as products and coproducts of other kinds of endomorphisms. An open problem: Apply this to the theory of reloids. An other open problem: Whether the category described in the above mentioned article is cartesian closed.