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.
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 […]
Principal staroids – an error corrected
I have uploaded a new version of my book with an error corrected: the definition of principal staroids and some related stuff were wrong.
An informal “conjecture”
Conjecture: Every time when an unproved result seems obvious, we refer to an unproved conjecture (from which it follows) in our mind
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 […]
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?
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 $latex \mathrm{Cor}\, \mathcal{A} = \uparrow^{\mathrm{Base} ( \mathcal{A})} \bigcap \mathcal{A}$.