A different definition of product of funcoids
Definition $latex a \mathrel{\left[ \prod^{(A 2)} f \right]} b \Leftrightarrow \exists M \in \mathrm{fin} \forall i \in (\mathrm{dom}\, f) \setminus M : \Pr^{\mathsf{RLD}}_i a \mathrel{[f_i]} \Pr^{\mathsf{RLD}}_i b$ for an indexed family $latex f$ of funcoids and atomic reloids $latex a$ and $latex b$ of suitable form. Here $latex M \in \mathrm{fin}$ means that $latex M$ […]
A conjecture proved
I’ve found an easy positive proof of this my conjecture.
Partial order and micronization (conjecture)
Conjecture $latex S^{\ast}(\mu(E)) = E$ for every partial order $latex E$. The function $latex S^\ast$ and micronization $latex \mu$ are defined in my research monograph.
A conjecture about outward funcoids
I’ve added to my book the following conjecture: Conjecture For every composable funcoids $latex f$ and $latex g$ $latex (\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq (\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}} f.$
I’ve solved a conjecture about pseudodifference of filters
I claimed earlier that I partially solved this open problem. Today I solved it completely. The proof is available in this PDF file.
Two new chapters in my math draft
I’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft. Both chapters are very rough draft and present not rigorous proofs but rough ideas.
Common generalizations of convergences and funcoids
I am reading the book “Convergence Foundations of Topology” by Szymon Dolecki and Frédéric Mynard. (Well, I am more skimming than reading, I may read it more carefully in the future.) After reading about a half of the book, I tried to integrate my theory of funcoids with their theory of convergences. And I noticed, […]
A funcoid related to directed topological spaces
The following problem arose from my attempt to re-express directed topological spaces in terms of funcoids. Conjecture Let $latex R$ be the complete funcoid corresponding to the usual topology on extended real line $latex [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\}$. Let $latex \geq$ be the order on this set. Then $latex R\sqcap^{\mathsf{FCD}}\mathord{\geq}$ is a complete funcoid.
Two equivalent conjectures
I have added to my book a short proof that the following two conjectures are equivalent: Conjecture $latex \mathrm{Compl}\,f \sqcap \mathrm{Compl}\,g = \mathrm{Compl}(f\sqcap g)$ for every reloids $latex f$ and $latex g$. Conjecture Meet of every two complete reloids is complete.
A new conjecture
While writing my book I overlooked to consider the following statement: Conjecture $latex f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S$ for principal funcoid $latex f$ and a set $latex S$ of funcoids of appropriate sources and destinations.