Overlapping subset neighborhoods and applications via generalized approximation spaces

In rough set theory, numerous studies have employed various neighborhood systems to develop novel rough approximation models, aiming to enhance accuracy and preserve as many characteristics as possible of the reference approximation space established by Pawlak. To contribute to this field, we introduce new rough neighborhoods, referred to as overlapping subset rough neighborhoods. They are defined under any arbitrary binary relation using subset relations between right and left neighborhoods, as well as between minimal right and minimal left neighborhoods. We examine their fundamental properties and identify the types of binary relations (i.e., serial, inverse serial, transitive, and quasi-order relations) under which they satisfy specific interrelations between them, as well as their relationships to previously established neighborhoods. We also successfully derive indicators based on the proposed neighborhoods to determine whether a relation is symmetric or both symmetric and transitive. Furthermore, we explore the relationships between the proposed neighborhoods as they transition between two generalized approx-imation spaces, where the smaller relation is serial (inverse serial) and transitive. Next, we put forward novel rough set models derived from the suggested neighborhoods and examine their core properties. We demonstrate that these models retain most of the characterizations of Pawlak approximation space and identify the binary relations under which they outperform previous models in minimizing uncertainty and satisfy the monotonicity property. Additionally, we design an algorithm to compute the boundary area and accuracy measures and provide a practical example related to book authorship to highlight the effectiveness of the proposed technique in extracting information. Finally, we conduct a comparative analysis that showcases the main advantages of our method while also referring to its limitations in comparison to various prior approaches.