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 […]
Products in dagger categories – updated
I have rewritten my draft article Products in dagger categories with complete ordered Mor-sets. Now I denote the product of an indexed family $latex X$ of objects as $latex \prod^{(Q)} X$ (instead of old confusing $latex Z’$ and $latex Z”$ notation) and the infimum product and supremum coproduct correspondingly as $latex \prod^{(L)} X$ and $latex […]
A failed attempt to prove a theorem
I have claimed that I have proved this theorem: Theorem Let $latex f$ is a $latex T_1$-separable (the same as $latex T_2$ for symmetric transitive) compact funcoid and $latex g$ is an reflexive, symmetric, and transitive endoreloid such that $latex ( \mathsf{FCD}) g = f$. Then $latex g = \langle f \times f \rangle \uparrow^{\mathsf{RLD}} […]