A draft about multifuncoids
I put online a rough preliminary draft about multifuncoids, a generalization of funcoids. It contains a few definitions, and theorems. Probably the most interesting thing in it is what I call graph-composition of multifuncoids. The draft contains several open problems.
“Upgrading a Multifuncoid” article upgraded
Now my article Upgrading a Multifuncoid is updated. The main change is that it now contains the conjecture “Upgrading a completary multifuncoid is a completary multifuncoid” (see the article for an exact formulation).
Product of two funcoids and product of two funcoids
I’ve put online an article (PDF, a partial draft) where I define product of two morphisms for certain categories. (Such products are pointfree funcoids.) Particularly it is defined product of two funcoids and product of two reloids. It is a more mature version of a draft I put online previously.
Some new theorems
I added the following theorems to Funcoids and Reloids article. The theorems are simple to prove but are surprising, as do something similar to inverting a binary relation which is generally neither monovalued nor injective. Proposition Let $latex f$, $latex g$, $latex h$ are binary relations. Then $latex g \circ f \not\asymp h \Leftrightarrow g […]
Micronization – the first attempt to define
This is my first attempt to define micronization. Definition Let $latex f$ is a binary relation between sets $latex A$ and $latex B$. micronization $latex \mu (f)$ of $latex f$ is the complete funcoid defined by the formula (for every $latex x \in A$) $latex \left\langle \mu (f) \right\rangle \left\{ x \right\} = \bigcap \left\{ […]
Product funcoids – a first messy draft
Product funcoids [outdated link remove] (not a math article but a messy collection of unproved and not exactly formulated statements). This is my first attempt to define product funcoids. There is needed yet much work to rewrite it as a rigorous math text.
Orderings of filters in terms of reloids, draft
I updated the article Orderings of filters in terms of reloids from “preliminary draft” to just “draft”. It means most errors are corrected and now you can read it.
The first problem in the chain is solved
I solved the first problem from this blog post (see Funcoids and Reloids article for a solution). It opens the path for solving several other open problems which seem to be its consequences.
Path for solving my open problems
I will outline which open problems follow from other open problems. In this post I don’t enter into gory details how to prove these implications, because these are useless without a prior proof of the main premise. I write these notes just not to be forgotten. It seems that from the first conjecture here follows […]
Two new conjectures
Conjecture If $latex a\times^{\mathsf{RLD}} b\subseteq(\mathsf{RLD})_{\mathrm{in}} f$ then $latex a\times^{\mathsf{FCD}} b\subseteq f$ for every funcoid $latex f$ and atomic f.o. $latex a$ and $latex b$ on the source and destination of $latex f$ correspondingly. A stronger conjecture: Conjecture If $latex \mathcal{A}\times^{\mathsf{RLD}} \mathcal{B}\subseteq(\mathsf{RLD})_{\mathrm{in}} f$ then $latex \mathcal{A}\times^{\mathsf{FCD}} \mathcal{B}\subseteq f$ for every funcoid $latex f$ and $latex \mathcal{A}\in\mathfrak{F}(\mathrm{Src}\,f)$, […]