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“Axiomatic Theory of Formulas: Algebraic Theory of Formulas”:

Mathematicians studied everything except formulas 🙂 Now there is an abstract axiomatic system for formulas, too. Read my book. For mathematicians, programmers, and students.

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Continuing this blog post: The set of all pointfree funcoids on upper semilattices with least elements is exactly a certain algebraic structure defined by propositional formulas.

Really just add the identities defining a pointfree funcoid to the identities of an upper semilattice with least element.

I will list the exact list of identities defining a pointfree funcoid on a poset with least element:

• $x \leq x$
• $x \leq y \land y \leq z \Rightarrow x \leq z$
• $x \leq y \land y \leq x \Rightarrow x=y$
• $\bot \leq y$
• $\lnot(\bot [f] y)$
• $\lnot(x [f] \bot)$
• $x \cup y \leq z \Leftrightarrow x \leq z \land y \leq z$
• $i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
• $k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$

(Here $[f]$ is a single relational symbol.)

This can be even generalized for upper semilattices with no requirement to have the least element, if we allow the least element constant symbol to be a partial function.

Note that for the general case of a pointfree funcoid between two upper semilattices, we have a partial algebraic structure because the operations are different for each of the two upper semilattices.

I am curious what are applications of this curious fact.

A few seconds ago I realized that certain cases of pointfree funcoids can be described as a structure in the sense of mathematical logic, that is as a finite set of operations and relational symbols.

Precisely, if a pointfree funcoid $f$ is defined on a lattice (or semilattice) with a least element $\bot$, then because a lattice (semilattice) is a relational structure defined by a set of propositional formula, then we have the following operations and relations:

• relation $[f]$
• meet and join operations on our lattice
• the least element of the lattice (a constant symbol)

and identities:

• the standard identities of lattice considered as an algebraic structure (and also the obvious identities for the least element)
• $\lnot(\bot [f] y)$
• $\lnot(x [f] \bot)$
• $i\cup j[f]k \Leftrightarrow i[f]k\lor j[f]k$
• $k[f]i\cup j \Leftrightarrow k[f]i\lor k[f]j$

What are applications of the curious fact that every funcoid is a structure defined by propositional formulas? I don’t yet know.

In recent days I published three scientific books on Amazon Kindle Direct Publishing Program.

My books contain complex formulas and yet I succeeded after much trying to find a way to publish them both as paperback and as e-books, yes with formulas. I want you to know how to do it, too.

## The Statistics or Why We Don’t Need Scientific Journals Anymore

Accordingly this listing, the world highest impact factor (more exactly, highest some “SJR” metrics) math journal is Journal of the American Mathematical Society with 3.93 cites per document (is it cites for two years or for 1 year measured during two years? unsure). There are journals with even higher cites (Mathematical Programming, Series B is 4.20 per document), but let consider this JAMS as the world most prestigious math journal (as it’s much above for example more famous Mathematische Annalen in this list).

Does it make sense to publish in this journal, yes, even this, world highest impact math journal, not speaking about weaker journals?

Consider further statistics: Accordingly this graph published at researchtrends.com, readers is just 0.59 of citations. (Yes, people cite more often than they read :-).)

3.93 x 0.59 = 2.31 readers per article of this journal per year.

Its tears. My e-book I published less than 30 days ago was purchased 3 times on Amazon last month (not last year!) So the conclusion: Self-publishing on Amazon one gets far more readers than even world most prestigious scientific journals.

Somebody may claim that my readership quality is bad. Surely, no: No housewife or carpenter would buy my e-book titled Algebraic General Topology. Volume 1, ever. So my readers are mathematicians.

Well, if your research field is really specialized, Amazon may be worse for you. There is no specialized topics like “Relationships between funcoids and reloids” on Amazon. (Just kidding, there is no such peer reviewed journal, too.) But if your topic is as wide as calculus or general topology, Amazon is perfect for you.

## Advantages of Amazon

Why to publish in a prestigious journal? Publishing instead on Amazon we have several advantages:

• You publish whatever you want, with no scientific reviewers.
• Review time is max 72 hours (really much faster).
• You get more readers, maybe tens times more!
• You can set your price to zero (self-made open access) or non-zero (the author gets the money, not our enemies!)
• You can even set the price to non-zero initially and change it to zero any time later, if you happen to have enough money.
• The maximum allowed e-book price at Amazon is $9.99, several times lower than big publishers want from poor people of the future. For paper book, you set any price you want (starting from as low as about$2.15 if you do not need profit).
• You can (especially if your price is non-zero) set a Web site and advertise it with SEO, mailing lists, etc. You sell more books while possibly getting profit.

I know exactly one case if you better to choose an official journal:

• If your religion happened to be the same as the religion of your government and therefore it wants to pay you money called “professor title”.

Otherwise publish on Amazon. Nothing prevents you to publish at arXiv, too.

## Technical Details of How to Publish

It remains only to explain how to accomplish publication technically, and it’s my main trick I’ve learned recently:

If your article is MS Word or LibreOffice format as this my book, it’s very easy: publish your .doc file. No need to explain how to use Kindle Direct Publishing, as it’s quite easy.

This way you publish both a paperback and a e-book.

But what if you need complex formulas that Word would not allow?

You have two options:

1. Use TeXmacs. It is a wonderful scriptable program that allows to write complex scientific texts without LaTeX. After you finished to write your text, convert it HTML. I was forced to use the mode with “formulas as images” (see TeXmacs preferences), because otherwise it once generated an empty file for me instead of a HTML file 😦 – even if the created file is non-empty, the formulas display wrongly in the Kindle reader. So just use the mode with “formulas as images”. When saving HTML with images, don’t forget to create an empty directory, otherwise the .png files pollute some existing directory on your computer. Also note that for a long time TeXmacs HTML export may take quite much time. Configure TeXmacs (“Edit” / “Preferences” / “Convert”) to export HTML (“Html” tab) with formulas as images and set image resolution (“Image” tab) in conversion to 300 DPI (300 DPI is an Amazon requirement). Next chdir to that directory (required!) and run a command (so it looks on Unix) like this:
mkdir -p tmp/limit texmacs -c article.tm tmp/limit/index.html -q
(cd tmp/limit && pandoc -f html -t epub -o ../../article.epub index.html)
ebook-polish -i -u article.epub article_polished.epub

Now you have a EPUB file article_polished.epub. Create the paper book in the usual way (upload a PDF file), but upload this .epub for the Kindle e-book.

Voila! You have a ebook with formulas. The formatting is not quite perfect but readable. An important note: Don’t use long (TeXmacs or LaTeX) inline formulas, they are not split into several lines in the resulting .epub but instead decrease in size if not fit page width, making it hard to read. You can use multline or align or whatever AMSMath formula environments instead.

Some notes:

• ebook-polish -i -u decreases the file size (and thus Amazon delivery cost).
• The resulting file article_polished.epub had mostly right (not too small, like it was when I used the original article.epub instead of article_polished.epub) formula sizes. I am not sure why: Probably, ebook-polish -i -u by an incident corrected the errors. Probably, Amazon changed their site software in the meantime. Anyway, it is not important why this last variant does work. It’s important that now we know a variant that does work.
• When I tried to use docx instead of epub, it produced a file unreadable with the Amazon site’s Kindle reader (apparently that reader has a bug). Not important for us now, as we can use .epub and ebook-polish to get good results.

2. You need to use LaTeX. Easy! Import the .tex file in TeXmacs (or even easier, replace in the commands above article.tm by article.tex, so simple) and next it is the same as the variant 1. Surprisingly, TeXmacs imports LaTeX files rather well (however, there are some problems: for example, some diagrams may be missing). There are some problems, but even my complex math book with my custom styles does work with only little glitches!

## Examples of publications

Now allow me to speak about my own publications done this way:

1. Algebraic General Topology. Volume 1. Edition 1 (paperback, e-book, homepage). Away with old bad topological spaces, we now have something better: funcoids and reloids (algebraic general topology generalization). Reads like a studybook for beginning students.
2. Axiomatic Theory of Formulas: Algebraic Theory of Formulas. Edition 1 (paperback, e-book, homepage). Mathematicians studied everything except of formulas. Now we have an axiomatic, algebraic theory of formulas. For mathematicians and programmers.
3. Limit of a Discontinuous Function: a Generalization for the Arbitrary Case (paperback, e-book, homepage). You know that there is no limit of discontinuous function. Just a moment, there is also no root of -1, right? There is likewise generalized limit of arbitrary function. Now we can easily define derivative of an arbitrary function and integral of arbitrary function! The values of limits of discontinuous functions are kinda infinities (“singularities”). I consider in this book also how to define differential equations with singular solutions. What is in the center of a black hole?

The last book in the above list is really just a short article. After choosing the smallest Amazon paper size, I made it 31 pages book while the smallest allowed book is 24 pages. So even shorter articles could not be published on Amazon 😦 Authors, just bundle.

Get down, scientific journals. Yahoo! We have Amazon.

A new book for mathematicians and programmers published: Axiomatic Theory of Formulas or Algebraic Theory of Formulas. The book is an undergraduate level but contains a new theory.

Get it:

From the preface:

This new mathematical theory developed by the book author researches the properties of mathematical formulas (aka expressions). Naturally this theory is essential for the mathematics.

Formulas are encountered in all areas of mathematics. Probably of special interest are propositional formulas in math logic. Using Operator Theory of Formulas may probably simplify proofs of math logic theorems.

The Theory of Formulas is a new foundation for math logic which is the foundation of mathematics. That is the Theory of Formulas is the foundation of the foundation of mathematics.

This theory researches mathematical formulas as well as any other objects which have constituent parts of different kinds (without the limit that a whole cannot be a part of its part, that is loops are not disallowed). For example, consider relations between objects in the memory of a computer or components of an electronic schema connected with different kinds of links.

The Operator Theory of Formulas is not limited only to finite neither only to acyclic formulas. Infinite and cyclic (e.g. a(b(a(b(…))))) formulas are also modeled by this theory. A common example of a system of infinite formulas is the set of irrational numbers. Another example of an infinite formula may be the entire mathematics in the form of some formalism (e.g. all groups of Group Theory).

This book contains a new axiomatic theory, the theory of constructs, and its specializations, including the theory of formulas:

• Informally, a construct is anything what has indexed components (parts). E.g., the a-(b+c) can be considered as a construct which has “a” as the part with index 1, “-” as the part with index 2, and (b+c) (a subformula) as the part with index 3.
• Formulas is a particular case of constructs. Informally, a formula is a construct which contains symbols. E.g. a-(b+c) would be a formula which contains symbols such as a, b, c, +, -.

To increase flexibility and power of the theory I study operators (functions) on the sets of constructs rather than the constructs themselves. So different kinds of constructs can be researched with a uniform method.

An important aspect of the theory of formulas accordingly this book is so called specialization that is adding additional subexpressions to expressions. For example if + would mean an abelian group operation, +1 (+ with added additional subformula “1”) would mean e.g. addition of whole numbers which is a specialization of abelian group operation etc. Specialization can be used instead of variable substitution.

Recall that formalist philosophy is a school in philosophy of mathematics which considers mathematical formulas as physical objects. For this new theory of formulas which includes infinite logical formulas that philosophy (“formalism”) is not applicable as infinite formulas cannot be written and are not physical objects in this sense. So this theory also changes the philosophy of mathematics.

### Draft Status

This is a preliminary version. If you will find any mistakes, please contact me.

Algebraic General Topology (a book series for both postdoctorals and college students) is a new branch of mathematics that replaces General Topology. Yes, general topology is now legacy! We have something better than topological spaces, funcoids. You almost spent time in vain studying topological spaces: In not so far future colleges will teach funcoids courses instead of topological spaces courses, topological spaces will remain only in very specialized courses. Funcoids behave better than topological spaces.

The author of the book managed to formulate general topology (continuity, separability, convergence/limit, connectedness, total boundness, etc.) in terms of algebraic formulas. It is done without using topological spaces (however, topological spaces are also considered in the book) using the author’s concept of funcoids and reloids.

Funcoids and reloids replace and generalize all of the following:

• topological spaces
• pretopological spaces
• proximity spaces
• uniform spaces
• even (directed) graphs

Yes, properties of topologies and graphs are described by the same formulas! We have a common generalization of topology/calculus and discrete mathematics.

For example, this is a formula (in the book there are three!) of all kinds (“regular”, uniform, proximal, discrete, others) continuity: fabf. Here f may be a function and a and b are spaces (be it topological, uniform, etc.) between which the function acts. In fact, continuity is defined for every partially ordered (pre)category.

For a further surprise, there is a formula for (generalized) limit of arbitrary (discontinuous) function:

lim f = { ν ∘ fr | rG }.

Now in the book’s system every function (between a wide class of spaces) is differentiable, every integral can be taken. And these limit do behave well: For example, if y is a value of the generalized limit, then yy = 0. Just open your mouth and try to realize the revolution that expects calculus soon.

Before going to the author’s discoveries, the book teaches the basic order, category, group theory and some of the legacy general topology, in order to be readable by anyone who knows basic set theory (and basic calculus to understand what the book is about).

After this the book presents some minor new results on order theory.

Then the author goes to the topic of filters on sets, lattices, and partially ordered sets. No doubt, this book is the world best reference on the topic of filters. Moreover, the author does not stop on the topic of filters on posets, but considers their generalization, filtrators. Filtrators is a very simple (it is just a pair of a poset and its subset) but powerful concept: most of the properties of filters do generalize for filtrators. The book reveals many previously unknown things about filters.

Then it starts the most interesting thing in the book: the theory of funcoids. Starting with an informal introduction, the author then considers funcoids in deep. It appears that funcoids are simultaneously a generalization of topological spaces, pretopological spaces, proximity spaces, (directed) graphs. The usual theory of topological spaces included but in the more general form of funcoids instead of spaces.

Then it follows the theory of reloids. Reloids is a very simple thing: a reloid is a filter on a Cartesian product of two sets. This is a generalization of the well known concept of uniform space. As you may know, uniform spaces describe such things as uniform continuity and total boundness on metric spaces.

The most interesting thing with funcoids and reloids is that they form a kind of algebra. So the name algebraic general topology. I have already shown you the formula (I remind: one of three formulas) of continuity. The author says that his main intention was to clean the mess: general topology was a mess of formulas with quantifiers where everybody could be lost, now it is instead an algebra, a beautiful theory.

The next chapter of the book considers interrelations between funcoids and reloids. And there are surprises.

The book considers even pointfree topology (topology without “points” or “numbers”). But not frames and locales, but pointfree funcoids instead, a simple easy generalization of funcoids.

The next thing in the book is kinda “multidimensional” general topology. The traditional point-set topology was kinda 2-dimensional, the author considers the arbitrary infinite dimensional topology (and yes, it does have applications, as applying multiple argument functions to limits of discontinuous functions needs this knowledge). And this infinite dimensional topology is also pointfree in the book.

Finally, the author presents a fascinating life story of the discovery. The main formula was discovered on the streets by a hungry homeless… This simple formula could have been discovered in 1937 but nobody except of a homeless first-year college student (the author) was able to guess it. Funcoids can be defined by four axioms that are easier than axioms of group theory, and nobody was able to guess.

Buy the book now (link in the beginning of this blog post), to jump to the very frontiers of the math research, whether you are a postdoctoral or a first-year student of a college.

If you are a teacher, you can make the following college courses using it as a studybook:

• basic order theory
• (co-)brouwerian lattices
• filters and filtrators
• funcoids
• reloids
• interrelationships between funcoids and reloids
• multidimensional general topology
• and more

Donate for this math research!

No root of -1? No limit of discontinuous function?

Like as once roots were generalized for negative numbers, I succeeded to generalize limits for arbitrary discontinuous functions.

The formula of limit of discontinuous function is based on algebraic general topology, my generalization of general topology in an algebraic way.

The formula that defines limit of discontinuous function is surprisingly simple:

lim f = { ν ∘ fr | rG }.

(This formula can be enhanced in different ways to make it behave better algebraically, but the idea is this.) And yes, it is very good algebraically, for example yy = 0 as if it were just a real number!

So we can for example, define derivative of an arbitrary real function. It opens a way to wholly new discontinuous calculus.

There is a problem however: I didn’t yet succeeded to put this generalized derivative into a differential equation, because the left and the right parts of the equation would belong to different sets and we could not simply equate them. In my opinion, it is probably the most important current problem in all mathematics to invent a way to put my derivative into a differential equation.

What will happen next? I don’t know, but maybe for example, we will discover what is the structure at the point of singularity in a black hole.