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…
read moreUsing this recently proved theorem I have defined product and co-product of endo-reloids. It is expressed by elegant algebraic formulas.
read moreI’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 $latex G$…
read moreI 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….
read moreI 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….
read moreRecently 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…
read moreComposition 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…
read moreIn 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…
read moreI’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…
read moreLet 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…
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