A new conjecture about relationships of funcoids and reloids
Conjecture $latex (\mathsf{RLD})_{\mathrm{in}} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)}\, f$ for every funcoid $latex f$. (I use notation from this note and this draft article.)
Funcoids as filters and composition
I have recently proved that there is an order isomorphism between funcoids and filters on the lattice of finite unions of Cartesian products of sets. Today I’ve proved that this bijection preserves composition. See this note (updated) for the proofs.
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 conjecture – finally solved
I have published online a short article saying that the set of funcoids is isomorphic to the set of filters on a certain lattice. Then I found a counter-example and decided that my theorem was wrong. I was somehow sad about this. But now I’ve realized that the counter-example is wrong. So we can celebrate […]
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 […]
I am going to rewrite my research monograph
When my book was already sent to a publisher, I decided to rewrite it. Here is my rewriting plan (what I am going to change in the book).
Four sets equivalent to filters on a poset
In this short note I describe four sets (including the set of filters itself) which bijectively correspond to the set of filters on a poset. I raise the question: How to denote all these four posets and their principal elements? Please write to my email any ideas about this.
A simple theorem not noticed before
Today I have noticed a new simple to prove theorem which is missing in my article: Theorem If $latex ( \mathfrak{A}; \mathfrak{Z})$ is a join-closed filtrator and both $latex \mathfrak{A}$ and $latex \mathfrak{Z}$ are complete lattices, then for every $latex S \in \mathscr{P} \mathfrak{A}$ $latex \mathrm{Cor}’ \bigsqcap^{\mathfrak{A}} S = \bigsqcap^{\mathfrak{Z}} \langle \mathrm{Cor}’ \rangle S.$ Particularly […]
Little changes in my math book
The section “Filters on a Set” of the preprint of my math book is rewritten noting the fact that $latex \mathrm{Cor}\, \mathcal{A} = \uparrow^{\mathrm{Base} ( \mathcal{A})} \bigcap \mathcal{A}$.