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Product funcoids [outdated link remove] (not a math article but a messy collection of unproved and not exactly formulated statements).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 $latex \mathcal{A}\in\mathfrak{F}(\mathrm{Src}\,f)$, […]
A new conjecture about funcoids and reloids
I’ve forgotten this conjecture when wrote Funcoids and Reloids article: Conjecture $latex (\mathsf{RLD})_{\mathrm{in}} (g\circ f) = (\mathsf{RLD})_{\mathrm{in}} g\circ(\mathsf{RLD})_{\mathrm{in}} f$ for every composable funcoids $latex f$ and $latex g$. Now this important conjecture is in its place in the article. I am going also to spend some time attempting to prove it.
The article “Conjecture: Upgrading a multifuncoid”
I first formulated the conjecture about upgrading a multifuncoid in this blog post. Now I’ve put online an article which is essentially the blog post with added proofs for the cases of n=0,1,2, converted into PDF format. (The conjecture is open for the case n=3 and above.) The terms multifuncoid and upgrading are defined in […]
A new conjecture about funcoids
Conjecture For every composable funcoids $latex f$ and $latex g$ we have $latex g \circ f = \bigcap \{ \uparrow^{\mathsf{FCD} ( \mathrm{Src}\,f ; \mathrm{Dst}\,g) } ( G \circ F ) \hspace{0.5em} | \hspace{0.5em} F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$.