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 the same word “generalization”? What is common in these two? Please comment.

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**Statement** For every composable funcoids and we have

The counterexample is and , where is an arbitrary nontrivial ultrafilter and 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.

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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 But my book is not peer reviewed.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!

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Note that as it is easy to prove for every set 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.

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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.

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**Theorem** Let be a distributive lattice with least element. Let . If exists, then also exists and .

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

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

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I knew that composition of two complete funcoids is complete. But now I’ve found that for to be complete it’s enough to be complete.

The proof which I missed for years is rather trivial:

Thus is complete.

I will amend my book when (sic!) it will be complete.

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**Proposition** For , a finite set and a function there exists (obviously unique) such that for and for .

This funcoid is determined by the formula

and its corollary:

**Corollary** If , , and , then there exists an (obviously unique) such that for all ultrafilters except of and .

This funcoid is determined by the formula

.

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