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Theory of Dependencies (New Theory Around Categories and Algebras)

Announces of my publications and the current state of the research

Journal of post-Axiomatic Mathematics and Logic - I started this online journal.

During my research of mathematical formulas (expressions) I introduced new concept dependency. Researching of dependencies appeared to be interesting by itself.

So, I introduce a new branch of mathematics, the theory of dependencies.

A dependency is simply a multidimensional (multi-argument) relation with two special variables X and Y (this is called dependency from X to Y). The simplest example of a dependency is a binary relation (e.g. a function).

Like binary relations, dependencies can be composed, inversed, etc.

The most interesting thing is that dependencies form several related categories (in the sense of category theory).

As of August 1 of 2005 I have not yet strictly proved this but it seems that:

theory of dependencies is a generalization if universal algebra (algebraization of algebra), in this role being an intermediary between universal algebra and category theory;
in an another role, theory of dependencies is a generalization of category theory (algebraization of category theory, it can be said that theory of dependencies it the algebra of categories).

Any systems of formulas (expressions) are described as a particular case of theory of dependencies (a system of expressions can be considered as a ternary relation).

To read the following text currently MathML supporting browser is required.

Theory of Dependencies

Entire dependencies article as one page By parts:

My newer generalizations of the theory of dependencies

Category of endomorphisms and pseudomorphisms - my new generalization of the theory of dependencies as a category induced by an arbitrary category with partially ordered morphisms.
Theory of Dependencies (rough draft) - new version of dependencies theory article.
Axiomatic theory of dependencies (partial draft) - my earlier (not quite successful) attempt to reformulate the theory of dependencies for general case of semigroups.

Theory of Constructs (Dependencies with Parameter)

Entire Systems of Constructs article as one page By parts:

Definition of Systems of Constructs
Repeatedly Applied Dependencies (Relations with Parameters)
Disjunctive Pseudomorphisms (an important specific case of pseudomorphisms)
Factor of Two Constructs (function which transforms one formula to an other)

The above drafts are already quite readable. See theory of formulas for a continuation of these online texts.

See also Modeling categories using dependencies, a possible way of algebraization of category theory.

Theory of Dependencies (New Theory Around Categories and Algebras)

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