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.
New concept: metamonovalued morphisms
Let fix some dagger category every of Hom-sets of which is a complete lattice, and the dagger functor agrees with the lattice order. I define a morphism $latex f$ to be monovalued when $latex \circ f^{-1}\le \mathrm{id}_{\mathrm{Dst}\, f}$. I call a morphism $latex f$ metamonovalued when $latex (\bigwedge G) \circ f = \bigwedge_{g \in G} […]
Crucial error in my definition of product funcoids
I have noticed a crucial error in the article with my definition of product funcoids (I confused the direction of an implication). Thus it is yet not proved that my “product” is really a categorical product. The same applies to my definition of coproduct.