Algebraic, Topological, and Mereological Foundations of Existential Granules

TL;DR

This paper introduces new algebraic, topological, and mereological concepts of existential granules that determine themselves and interact with their environment, aiming to enhance granular computing and classification methods.

Contribution

It invents and characterizes self-determining existential granules from multiple perspectives, integrating them into theoretical frameworks and applications in rough sets and soft computing.

Findings

Existential granules are characterized across algebraic, topological, and mereological frameworks.

They are shown to fit into various theoretical models of granular computing.

The paper proposes algorithms and applications for classification problems.

Abstract

In this research, new concepts of existential granules that determine themselves are invented, and are characterized from algebraic, topological, and mereological perspectives. Existential granules are those that determine themselves initially, and interact with their environment subsequently. Examples of the concept, such as those of granular balls, though inadequately defined, algorithmically established, and insufficiently theorized in earlier works by others, are already used in applications of rough sets and soft computing. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations of the approach. Additionally, many open problems are posed and directions provided.