Overlapping containment rough neighborhoods and their generalized approximation spaces with applications

In rough set theory, we distinguish confirmed and possible data, extracted through subsets utilizing lower and upper approximations, respectively. Earlier studies have presented several rough approximation models, drawing inspiration from neighborhood systems, aimed at enhancing accuracy degree and satisfying the axioms of standard approximation space, introduced by Pawlak. In this article, we first introduce novel rough neighborhoods so-called overlapping containment rough neighborhoods, denoted by \(\widetilde{\textrm{C}}_k\) ${\tilde{\text{C}}}_k$ -neighborhoods, using inclusion relations between \(\mathcal {D}_r\) ${\mathcal{D}}_r$ -neighborhoods and \(\mathcal {D}_l\) ${\mathcal{D}}_l$ -neighborhoods, as well as \(\mathcal {D}_{\langle r\rangle }\) ${\mathcal{D}}_{⟨r⟩}$ -neighborhoods and \(\mathcal {D}_{\langle l\rangle }\) ${\mathcal{D}}_{⟨l⟩}$ -neighborhoods, all defined under an arbitrary relation. We explore their main characterizations and reveal the relationships between them under specific types of binary relations, such as symmetric, transitive, and partial order relations. As a unique contribution, we successfully derive an indicator inspired by \(\widetilde{\textrm{C}}_k\) ${\tilde{\text{C}}}_k$ -neighborhoods for \(k\in \{r,l,i\}\) $k\in \{r,l,i\}$ to determine whether a relation is symmetric. Additionally, we describe the behavior of \(\widetilde{\textrm{C}}_k\) ${\tilde{\text{C}}}_k$ -neighborhoods as they navigate between two generalized approximation spaces, where the relations are reflexive and transitive, and one is a subset of the other. Then, we exploit \(\widetilde{\textrm{C}}_k\) ${\tilde{\text{C}}}_k$ -neighborhoods to present fresh rough set models. We examine their main properties and demonstrate that they keep most characterizations of Pawlak’s paradigm while reducing the uncertainty in the data compared to some previous studies, also, we show that they satisfy the monstrosity property under quasi-order relations. To elucidate the superiority and accuracy of the present approach, we apply it to analyze the information systems related to the authorship of articles and books by selected authors and conduct a comparative analysis with several preceding approaches. Finally, a summary of the obtained results and relationships and suggestion for some forthcoming work are offered.