Ordered Semicategory Actions

General Topology as Ordered Semigroup Actions

Several new axiomatic systems as general and as important as group theory. It starts from generalizing proximity spaces and topologies and later moves to ordered semigroup actions and ordered semicategory actions that encompass all kinds of spaces that are met in general topology: from locales and frames to metric spaces: topological spaces, uniform spaces, proximity spaces, Cauchy spaces, and also for graphs and digraphs (directed graphs).

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The particular topics

In this work I introduce and study in details the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids. Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-differentiable solution of a (partial) differential equation. It’s raised the question how do such solutions “look like” starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx – \int_a^b f(x)dx = 0$ is defined and true for every function.

The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category. In one formula continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms.

I define space as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc. all are spaces. It can be further generalized to ordered semicategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.) in an arbitrary space. Now general topology is an algebraic theory.

Before going to topology, this book studies properties of co-brouwerian lattices and filters.

Purchasing this book, you support carbon accounting and DeSci (decentralized science).

Victor Porton

Your instructor is Victor Porton, the person who discovered ordered semigroup actions (and wrote 500 pages about them), a theory as general as group theory but unknown before. Victor Porton is a programming languages polyglot, author of multitudinous softwares and programming libraries, blockchain expert and winner of multitudinous blockchain hackathons, author of several books, a philosopher.
Victor Porton studied math in a university 4.5 years, but didn't receive a degree because of discrimination.

Reviews

How does it relate to general topology?

This book describes a generalization of general topology, done in an algebraic way instead of non-algebraic mess of traditional general topology.

Ordered semigroup actions and ordered semicategory actions were discovered by Victor Porton in 2019. At the same time it was discovered that every kind of spaces met in general topology, from metric spaces to locales and frames are fully characterizable by an element of an ordered semigroup action (or a morphism of an ordered semicategory action). Thus, general topology is nothing other as the algebra of ordered semicategory actions.

So, you discovered space-in-general?

Almost, I discovered the general notion of space for purposes of general topology. “Geometric spaces” (like Euclidean spaces) don’t (yet) fit into this notion.

Why do you want to be nominated for Abel Prize for this book?

I want prize money, because I need funds for further research and publication.

I think, completely reducing all general topology (from metric spaces to locales and frames) to an algebraic concept (ordered semicategory actions) is quite worth Abel Prize.

Ordered Semicategory Actions

General Topology as Ordered Semigroup Actions

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The particular topics

Victor Porton

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