Generalized rough approximation spaces inspired by cardinality neighborhoods and ideals with application to dengue disease

This article aims to define four new kinds of rough set models based on cardinality neighborhoods and two ideals. The significance of these methods lies in their foundation on ideals, which serve as topological tools. Furthermore, the use of two ideals offers two perspectives instead of just one, thereby reducing the boundary region and increasing the accuracy, which is the primary objective of rough set theory. The concepts of lower and upper approximations based on ideals are presented for the four types. Additionally, we establish essential properties and results for these approximations and construct counterexamples to demonstrate how some of Pawlak's properties have dissipated in the proposed models. The relationships between the current and previous approximations are discussed, and algorithms to classify whether a subset is exact or rough are introduced. Furthermore, we demonstrate how one combination of ideals is applied to address rough paradigms from a topological perspective. Practically, we apply the proposed paradigms to dengue disease management and elucidate two key points: first, our models are distinguished compared to previous ones by retaining most properties of the original approximation operators proposed by Pawlak; and second, we identify which of the proposed models is better at increasing the accuracy of subsets. In conclusion, we debate the advantages of the suggested models and the motivations behind each type, while also highlighting some of their shortcomings.