Algebraic, Topological, and Mereological Foundations of Existential Granules
This foundational work addresses theoretical gaps in granular computing, potentially benefiting researchers in rough sets and soft computing, though it appears incremental in building on prior concepts.
The paper introduces new concepts of existential granules that determine themselves, characterizing them from algebraic, topological, and mereological perspectives to fit into granular computing frameworks. It aims to enable algorithm development and applications in classification, while posing open problems for generalization.
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.