Victor Porton's Math Weblog

I significantly updated online article Category of Endomorphisms and Pseudomorphisms. (Updates concern mainly projection of a morphism and being generalized for precategories rather than only categories.) It is a base for my theory of formulas.

I suggest to add it to the base math University program (see also here).

This is nearly the final draft, the state in which this article will be published in Journal of post-Axiomatic Mathematics and Logic (to be used as a base for other articles) and will go to preprint and/or paper publication (when will be converted from an XML format to LaTeX).

New online math journal Journal of post-Axiomatic Mathematics and Logic is expected to start its activity soon. The journal will publish materials related to definitional and functional methods in math logic intended to replace axiomatic method (20 century mathematics) with 21 century post-axiomatic mathematics, as well as research concerning theory of formulas and related issues (see here and here for details).

First articles for the journal are near to be ready for publication. Please donate for publication of this mathematical journal and development of PostScript/PDF conversion tools associated with XSLT stylesheets for math (which will be free available for use by other math journals and math Web sites) and for development of the stylesheets themselves. Articles are planned to be available free of charge.

www.mathematics21.org

My category theory terminology page is updated. The most important changes: fixed error in definition of a category, added notion of trans-endomorphism used to define a typed morphism in an elegant way.

I suggest to use my variant of category theory terminology instead of the standard category theory definitions as found in math encyclopedias.

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I published online my definitions of basic category theory terms (category, precategory, functor, prefunctor, morphism, etc.) with some simple theorems about them. My definitions differ of the standard encyclopedic definitions. I suggest to use my definitions for any serious math research instead of the standard definitions. (I use my definitions for http://www.mathematics21.org/category-theory.html">my category theory research.)

Additionally, I have elegantly defined a category based on a semigroup.

Please donate for revolutionary math research (both fundamental and practical). donate for 21 century mathematics!

Now accepting credit cards (VISA and MC) as well as checks, e-gold, MoneyBookers, etc.

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In this small article I define inverse image of a category by a function and analyse its simple properties.

It is used in category theory research.

Theorem. Let f and g are binary relations. Then the following statements are equivalent:

1. g = f|_{dom g}.
2. Exists a set K such that g = f|_K.
3. gg^-1 = fg^-1 and g is a subset of f.
4. gg^-1 = gf^-1 and g is a subset of f.
5. gg^-1 = fg^-1 and dom g is a subset of dom f.
6. gg^-1 = gf^-1 and dom g is a subset of dom f.

An other, a little less convenient for usage but much easier to remember (mnemonic), equivalent form of this theorem is:

Theorem (mnemonic form). Given that f and g are binary relations and dom g is a subset of dom f, right multiplying (composing) both sides of the equation g = f|_{dom g} with g^-1 produces a formula equivalent to this equation.

I have stated this theorem as a conjecture in this weblog post. Now I have proved it. (The proof is below.)

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Conjecture about binary relations:

Let f and g are binary relations, g is a subset of f. If gg^-1 = fg^-1, then exists a set K such that g = f|_K.

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I suggest to add Category of Endomorphisms to program of math and computer science faculties of Universities. It should be added as a sub-course of Category theory university course.

This is especially important for Math Logic and Computer Science students, and also important for algebraists.

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Category of Endomorphisms and Pseudomorphisms online article updated.

The article is about categories whose objects are endomorphisms of any given category. Particularly is considered the case of a category with partially ordered morphisms. It produces a rich, interesting theory.

The section Category of Elements and below is completely reworked. (There were serious errors, now they are corrected.)

Category of epimorphisms and pseudomorphims (online article) updated with several new theorems, corrections of mistakes, and some new concepts.

This article considers an interesting category whose objects are endomorphisms of an arbitrary category with possibly partially ordered morphisms.

It is used for my algebraic theory of formulas. It will be also used in other fields of mathematics.

Online article (yet partial) about morphisms (I call these pseudomorphisms) of endomorphisms of a category with partially ordered morphisms (my new theory).

Pseudomorphism f from an endomorphism U to an endomorphism V is defined by formula f U \subseteq V f.

This is a purely categorical generalization of some aspects of my theory of dependencies, the theory of multidimensional relations (considered as generalization of binary relations) with special parameters X and Y. This theory is important for algebraic theory of formulas (expressions).

Read article online: Category of Endomorphisms and Pseudomorphisms.

I have made certain math discoveries. Please help me to receive Abel Prize (an about $1000000 prize for mathematicians, similar to Nobel Prize).

I ask you to write certain letter to the prize committee. You can do it if either you yourself are a mathematician or you have a friend mathematician.

I have done a work and it should be rewarded, agree? So, please help me.

Significantly updated theory of dependencies (algebraic theory of formulas).

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I have shown that a category (from category theory) can be represented as a pair of a ternary relation and a partial binary operation. This is a possible way of algebraization of category theory. See here.

Note that I use my definition of category with multiple sources and destinations.

Updated my online article Algebraic Theory of Formulas.

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I updated my online article Algebraic Theory of Formulas.

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This short article introduces two modified definitions of category with multiple sources and destinations (from category theory from abstract mathematics).

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I updated my online article Theory of Dependencies (main updates about morphisms between elements and categories of elements, also about composition of dependencies).

Theory of Dependencies is new theory about multidimensional (multi-argument) relations with two special arguments X and Y. This theory is especially important for axiomatic theory of (infinite) mathematical formulas (expressions).

With the idea of categories of elements we can descent from abstractions to concrete things such as categories related with separate elements.

I expect that this theory will supercede both universal algebra and category theory.