Two kinds of generalization
I noticed that there are two different things in mathematics both referred as “generalization”. The first is like replacing real numbers with complex numbers, that is replacing a set in consideration with its superset. The second is like replacing a metric space with its topology, that is abstracting away some properties. Why are both called with […]
A counterexample to my conjecture
I’ve found a counterexample to the following conjecture: Statement For every composable funcoids $latex f$ and $latex g$ we have $latex H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$ The counterexample is $latex f=a\times^{\mathsf{FCD}} \{p\}$ and $latex g=\{p\}\times^{\mathsf{FCD}}a$, $latex H=1$ where $latex […]
Peer review for open access math books
We have a new kind of math publishing: Free books distributed through Internet. It is a new kind of mathematical culture. Some books of this kind appeared with daunting success. It has great advantages. It is how things should be done in modern times. But it miss an essential part of math tradition, peer review. For […]
Offtopic: A religious reason to do scientific research
See this my post in other blog for a religious reason to do scientific research.
A new funcoid discovered
It is easy to prove that the equation $latex \langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X$ (for principal filters $latex X$) defines a (unique) funcoid $latex \mathscr{A}$ which I call quasi-atoms funcoid. Note that as it is easy to prove $latex \langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y$ for every set $latex Y$ of ultrafilters. […]
Funcoidal groups and a curious theorem
I started to work on funcoidal groups (a generalization of topological groups). I defined it and promptly found a curious theorem. Not sure if this theorem has use for anything. See the definition and the “curious” proposition in this draft. Note that I work on another projects and may be not very active in researching […]
New simple theorem in my book
I added to my online research book the following theorem: Theorem Let $latex \mathfrak{A}$ be a distributive lattice with least element. Let $latex a,b\in\mathfrak{A}$. If $latex a\setminus b$ exists, then $latex a\setminus^* b$ also exists and $latex a\setminus^* b=a\setminus b$. The user quasi of Math.SE has helped me with the proof.
A wrong result
I’ve published in my blog a new theorem. The proof was with an error (see the previous edited post)!
A new unexpected result (ERROR!)
The below is wrong! The proof requires $latex \langle g^{-1}\rangle J$ to be a principal filter what does not necessarily hold. I knew that composition of two complete funcoids is complete. But now I’ve found that for $latex g\circ f$ to be complete it’s enough $latex f$ to be complete. The proof which I missed […]
A new easy proposition about funcoids
I have proved (see new version of my book) the following proposition. (It is basically a special case of my erroneous theorem which I proposed earlier.) Proposition For $latex f \in \mathsf{FCD} (A, B)$, a finite set $latex X \in \mathscr{P} A$ and a function $latex t \in \mathscr{F} (B)^X$ there exists (obviously unique) $latex […]