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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Algebraic General Topology, Volume 1, unpacks the foundations of algebraic general topology, introducing novel concepts and exploring their applications.
The focus lies on two key advancements: funcoids and reloids. Funcoids offer a broader perspective on proximity spaces, while reloids provide a refined understanding of generalized uniform spaces. Both concepts extend traditional binary relations by employing filters instead of sets within their domains and ranges. This allows them to act as a unifying framework, encompassing the fields of calculus and discrete mathematics through a shared foundation.
In Algebraic General Topology, author Victor Lvovich Porton establishes a novel definition of continuity, expressed through an elegant algebraic formula. This definition applies to various morphisms (including funcoids and reloids) within the context of partially ordered categories. Notably, it encompasses traditional continuity, proximity continuity, and uniform continuity within a single, unifying formula. Additionally, the concept of connectedness is introduced specifically for funcoids and reloids.
Also the book discusses kinda multidimensional topology, compared to just two dimensions of the conventional point-set topology where there were considered relations of just two objects: a point and a set. The book also examines various product constructions applicable to funcoids and other morphisms.
Before starting to discuss topology, author Victor Proton lays the groundwork by investigating the properties of co-Brouwerian lattices and of filters. This foundational knowledge equips the reader with a deeper understanding of the subsequent topological concepts presented.
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Victor Porton
Victor Lvovich Porton is a self-taught mathematician, researcher, author of multiple published books, and CEO of Zon Social Network, with a fervent passion for revolutionizing the field. Despite embarking on a 4.5-year mathematics program, his studies were unfortunately cut short due to discrimination. However, this setback did not deter his pursuit of knowledge. Learn More ..
Reviews
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.
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.
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.
