One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals and E_j‐neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from E_j‐neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of j -adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

This article appoints a novel model of rough set approximations (RSA), namely, rough set approximation models build on containment neighborhoods RSA (CRSA), that generalize the traditional notions of RSA and obtain valuable consequences by minifying the boundary areas. To justify this extension, it is integrated with the binary version of the honey badger optimization (HBO) algorithm as a feature selection (FS) approach. The main target of using this extension is to assess the quality of selected features. To evaluate the performance of BHBO based on CRSA, a set of ten datasets is used. In addition, the results of BHOB are compared with other well-known FS approaches. The results show the superiority of CRSA over the traditional RS approximations. In addition, they illustrate the high ability of BHBO to improve the classification accuracy overall the compared methods in terms of performance metrics.

Real-world problems often involve imprecision and uncertainty, presenting challenges across various domains such as engineering, artificial intelligence, social sciences, and medical sciences. To bridge this gap, Pawlak introduced classical rough set models, focusing on upper and lower approximations based on equivalence relations. However, these relations restrict the applicability of rough sets in many contexts. This paper explores new approaches to extending rough set theory by introducing "primal approximation spaces". Primal is defined as novel structures designed to generalize rough set approximations beyond the traditional methods. This paper proposes a new technique for generating rough approximations by using κ-neighborhoods and primal which allows for creating diverse supra-topologies and enhancing the flexibility of rough set models. Furthermore, we here introduce bi-primal approximation spaces, a new form of approximation that can be examined through two distinct methods, as a result, revealing their unique characteristics and relationships. The research underlines the practical applications of these new methods by providing a detailed case study that demonstrates their effectiveness in solving decision-making problems. Moreover, this study also compares the primal-based methods with existing approaches based on ideals, illustrating their distinctive advantages and limitations. Overall, this work offers a remarkable advancement in rough set theory by expanding its theoretical framework and practical applicability through the introduction of primal and bi-primal approximation spaces.

Topology is a beneficial structure to study the approximation operators in the rough set theory. In this work, we first introduce six new types of neighborhoods with respect to finite binary relations. We study their main properties and show under what conditions they are equivalent. Then we applied these types of neighborhoods to initiate some topological spaces that are utilized to define new types of rough set models. We compare these models and prove that the best accuracy measures are obtained in the cases of $i$ and $〈i〉$ . Also, we illustrate that our approaches are better than those defined under one arbitrary relation. To improve rough sets’ accuracy, we define some topological spaces using the idea of ideals. With the help of examples, we demonstrate that our methods are better than some methods studied in some published literature. Finally, we give a real-life application showing the merits of the approaches followed in this manuscript.