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.
Preliminary announce: n-ary funcoids
I am now developing a theory of generalization of funcoids, n-ary (multi)funcoids (where n is a set). Previously I was going to make first theory of finitary multifuncoids that is the case when n is finite, because there were some complexities with proving some important theorems about infinitary funcoids. Today I managed to prove it […]
Slides about Algebraic General Topology
I prepared PDF slides for quickly familiarizing a reader with Algebraic General Topology. I hope to give a talk with these slides at a math research conference. The current version of this PDF contains 54 slides.
Oblique products, related theorems and conjectures
I updated Funcoids and Reloids article. Now it contains a section on oblique products. It now contains also the following conjectures: Conjecture $latex \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B}$ for some f.o. $latex \mathcal{A}$, $latex \mathcal{B}$. Conjecture $latex \mathcal{A} \times^{\mathsf{RLD}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}$ for some f.o. $latex \mathcal{A}$, […]
Uniformization of funcoids
I’ve put a sketch draft of future research “Uniformization of funcoids” on my Algebraic General Topology page. “Uniformization” is meant to be a generalization of oblique products of filters. This future research may be published as a part of my Funcoids and Reloids article, or (what is more likely) as a separate article.
Oblique product of filters
Funcoids and reloids are my research in the field of general topology. Let $latex \mathcal{A}$ and $latex \mathcal{B}$ are filters. Earlier I introduced three kinds of products of filters: funcoidal product: $latex \mathcal{A}\times^{\mathsf{FCD}}\mathcal{B}$; reloidal product: $latex \mathcal{A}\times^{\mathsf{RLD}}\mathcal{B}$; second product: $latex \mathcal{A}\times^{\mathsf{RLD}}_F\mathcal{B}$; The last two products are reloids while the first is a funcoid. Funcoidal and […]