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.
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?
A bijection from the set of funcoids to a certain subset of reloids
I have proved that there is a bijection from the set $latex \mathsf{FCD}(A;B)$ to a certain subset of $latex \mathsf{RLD}(A;B)$ (which I call funcoidal reloids). See section (currently numbered 8.4) “Funcoidal reloids” in the preprint of my book.
“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} […]
Cauchy spaces and Cauchy continuity expressed through reloids
In the march of development of my theory, I have expressed many kinds of spaces (topological, proximity, uniform spaces, etc.) through funcoids and reloids subsuming their properties (such as continuity) for my algebraic operations. Now I have expressed Cauchy spaces (and some more general kinds of spaces) through reloids. And yes, Cauchy continuity appears as […]
I thought it was stolen :-)
When I first saw topogenous relations at first I thought that my definition of funcoids was plagiarized (for some special case). But then I looked the year of publication. It was 1963, long before discovery of funcoids. Topogenous relations are a trivial generalization of funcoids. However, I doubt whether anyone (except of myself) has defined […]
New theorem about funcoids and reloids
I’ve added to the preprint of my book a new theorem (currently numbered theorem 8.30). The theorem states: Theorem $latex g \circ ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) \circ f = \langle ( \mathsf{FCD}) f^{- 1} \rangle \mathcal{A} \times^{\mathsf{RLD}} \langle ( \mathsf{FCD}) g \rangle \mathcal{B}$ for every reloids $latex f$, $latex g$ and filters $latex \mathcal{A} \in […]
Cauchy filters, a generalization for reloids
I have proved the theorem from Wikipedia that every Cauchy filter is contained in a maximal Cauchy filter (in fact I’ve proved a more general statement). I don’t know the standard proof (and don’t know where to find it), so I’ve devised a proof myself. I hope (thanks to use the notation reloids in my […]