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Product funcoids [outdated link remove] (not a math article but a messy collection of unproved and not exactly formulated statements).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.
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 $latex g \circ f = \bigcap \{ \uparrow^{\mathsf{FCD} ( \mathrm{Src}\,f ; \mathrm{Dst}\,g) } ( G \circ F ) \hspace{0.5em} | \hspace{0.5em} F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \}$.
Conjecture: Upgrading a multifuncoid
This short article is the first my public writing where I introduce the concept of multidimensional funcoid which I am investigating now. But the main purpose of this article is to formulate a conjecture (see below). This is the shortest possible writing enough to explain my conjecture to every mathematician. Refer to this Web site […]
My research is stalled
My research of n-ary funcoids is stalled now, as I am (yet) unable to solve certain problem. I posted a special version of this problem to MathOverflow. Please help me to solve this open problem. It is a very important problem.