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?
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).
Error correction: definition of multifuncoids was wrong
I have corrected some errors in my book related to my definition of multifuncoids. The definition of multifuncoid was wrong and this triggered several errors along my book.
I added a new chapter to my book
After checking for errors I added (as a new chapter) materials of the article Identity staroids into my research monograph. I plan to “dissolve” this chapter, that is distribute its materials among other chapters and liquidate this chapter itself.
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 […]