I’ve found a counterexample to the following conjecture:

Statement For every composable funcoids $f$ and $g$ we have

$H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$

The counterexample is $f=a\times^{\mathsf{FCD}} \{p\}$ and $g=\{p\}\times^{\mathsf{FCD}}a$, $H=1$ where $a$ is an arbitrary nontrivial ultrafilter and $p$ is an arbitrary point.

I leave the proof that it is a counterexample as an easy exercise for the reader (however I am going to add the proof to my book soon).

That the conjecture failed invalidates my new proof of Urysohn’s lemma which was based on this conjecture.

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 articles there are open access journals. But for book form of dissemination of knowledge it is pitifully missing! And my prediction is that as math advances and becomes more of integration of sets of related ideas rather than just a proof of a particular theorem, book form of research will become more and more appropriate gradually replacing traditional article-based publishing. Thus we need it even more.

I self-published a book with my (seminal) novel research. The book is structured like a textbook and is accessible for beginning students. I put it (and even its LaTeX source) online for free. But my book is not peer reviewed. That’s a problem. I sent it to several traditional publishers, but they don’t want to put effort and money into a book which is already available for free.

The ultimate solution would be an academic publisher which would peer review free books. Ideally, they would do it for free.

I urge the mathematical community to invest resources into making such a new kind of publisher. This would integrate both novel Internet way and tradition of peer review. Please distribute this organization idea at conferences, meetings, math forums and blogs. It would be really great to implement this idea. We just need a will of an academy and some little money. It is worth to use these money for a benefit of the world as they would enhance development of science in the world. Do it!

See this my post in other blog for a religious reason to do scientific research.

It is easy to prove that the equation $\langle \mathscr{A} \rangle X = \mathrm{atoms}^{\mathfrak{A}}\, X$ (for principal filters $X$) defines a (unique) funcoid $\mathscr{A}$ which I call quasi-atoms funcoid.

Note that as it is easy to prove $\langle \mathscr{A}^{-1} \rangle Y = \bigsqcup Y$ for every set $Y$ of ultrafilters.

Does this funcoid posses interesting properties? Can it be used to prove any open problem?

What is its behavior on non-principal filters?

I started researching properties of this weird funcoid in this PDF file.

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 funcoidal groups in near time.

I added to my online research book the following theorem:

Theorem Let $\mathfrak{A}$ be a distributive lattice with least element. Let $a,b\in\mathfrak{A}$. If $a\setminus b$ exists, then $a\setminus^* b$ also exists and $a\setminus^* b=a\setminus b$.

The user quasi of Math.SE has helped me with the proof.

I’ve published in my blog a new theorem.

The proof was with an error (see the previous edited post)!