Correction on the recent theorems
About new theorems in in this my blog post: I’ve simplified this theorem: Theorem A reloid $latex f$ is complete iff $latex f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle […]
Some new theorems
I’ve proved some new theorems. The proofs are currently available in this PDF file. Theorem The set of funcoids is with separable core. Theorem The set of funcoids is with co-separable core. Theorem A funcoid $latex f$ is complete iff $latex f = \bigsqcap^{\mathsf{FCD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T […]
Funcoids are filters?
I am not doing math research this month (because a bug in TeXmacs software which I use for writing my book and articles). I instead do writing some free software not to waste my time. But today (this hour) I unexpectedly had a new interesting idea about my math research: Let denote $latex Q$ the […]
Draft article about identity staroids
I’ve mostly finished writing the article Identity Staroids which considers $latex n$-ary identity staroids (with possibly infinite $latex n$), which generalize $latex n$-ary identity relations and some related topics. (In my theory there are two kinds of identity staroids: big and small identity staroids.) Writing of the article is mostly finished, I am going just […]
Coatoms of the lattice of funcoids
Open problem Let $latex A$ and $latex B$ be infinite sets. Characterize the set of all coatoms of the lattice $latex \mathsf{FCD}(A;B)$ of funcoids from $latex A$ to $latex B$. Particularly, is this set empty? Is $latex \mathsf{FCD}(A;B)$ a coatomic lattice? coatomistic lattice?
“Funcoidal” reloids, a new research idea
Just today I’ve got the idea of the below conjecture: Definition I call funcoidal such reloid $latex \nu$ that $latex \mathcal{X} \times^{\mathsf{RLD}} \mathcal{Y} \not\asymp \nu \Rightarrow \\ \exists \mathcal{X}’ \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{X})} \setminus \{ 0 \}, \mathcal{Y}’ \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{Y})} \setminus \{ 0 \} : ( \mathcal{X}’ \sqsubseteq \mathcal{X} \wedge \mathcal{Y}’ \sqsubseteq \mathcal{Y} […]
A failed attempt to prove a theorem
I have claimed that I have proved this theorem: Theorem Let $latex f$ is a $latex T_1$-separable (the same as $latex T_2$ for symmetric transitive) compact funcoid and $latex g$ is an reflexive, symmetric, and transitive endoreloid such that $latex ( \mathsf{FCD}) g = f$. Then $latex g = \langle f \times f \rangle \uparrow^{\mathsf{RLD}} […]
New theorem about core part of funcoids and reloids
Today I’ve proved a new little theorem: Theorem $latex \mathrm{Cor} ( \mathsf{FCD}) g = ( \mathsf{FCD}) \mathrm{Cor}\, g$ for every reloid $latex g$. Conjecture For every funcoid $latex g$ $latex \mathrm{Cor} ( \mathsf{RLD})_{\mathrm{in}} g = ( \mathsf{RLD})_{\mathrm{in}} \mathrm{Cor}\, g$; $latex \mathrm{Cor} ( \mathsf{RLD})_{\mathrm{out}} g = ( \mathsf{RLD})_{\mathrm{out}} \mathrm{Cor}\, g$. See my book.
Conjecture: Connectedness in proximity spaces
I’ve asked this question at math.StackExchange.com Let $latex \delta$ be a proximity. A set $latex A$ is connected regarding $latex \delta$ iff $latex \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$. Conjecture Set $latex A$ is connected regarding $latex \delta$ iff for […]
Conjecture about funcoids proved
I’ve proved this my conjecture: $latex g \circ f = \bigsqcap \left\{ G \circ F \,|\, F \in \mathrm{up}\, f, G \in \mathrm{up}\, g \right\}$ for every composable funcoids $latex f$ and $latex g$. See my book (in the current draft the theorem 6.65) for a proof.