Two new theorems about complete reloids
I have just proven the following two new theorems: Theorem Composition of complete reloids is complete. Theorem $latex (\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f)$ if $latex f$ and $latex g$ are both complete funcoids (or both co-complete). See this note for the proofs.
Another (easy) new theorem
I’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions. So my new theorem: Theorem $latex (\mathsf{FCD})$ and $latex (\mathsf{RLD})_{\mathrm{out}}$ form mutually inverse bijections between complete reloids and complete funcoids. For a proof see this note.
A proposition about complete funcoids and reloids
In this recent blog post I have formulated the conjecture: Conjecture A funcoid $latex f$ is complete iff $latex f=(\mathsf{FCD}) g$ for a complete reloid $latex g$. This conjecture has not been living long, I have quickly proved it in this note.
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? Conjecture II
Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample. Now I propose another similar but weaker open problem: Conjecture Let $latex U$ be a set. The […]
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.