AI Summary & Key Takeaways
AI InsightMoreover, generalized nano topologies are constructed from these new structures, and their applicability is demonstrated through a real-world example.
📌 3 Key Takeaways
Abstract
In 1982, Pawlak introduced the concept of rough sets as a mathematical framework for addressing real-world problems. However, the dependence on equivalence relations within this framework has often proven inadequate for handling more complex cases. The aim of this study is to develop a more effective approach for such situations. In particular, partial order relations are combined with $\mathit{j}$-neighborhoods to capture not only vague similarities among objects but also hierarchical dependencies and priority structures that frequently arise in real data. For this purpose, ordered $\mathit{j}$-neighborhood classes are defined by employing increasing and decreasing sets based on partial order relations together with $\mathit{j}$-neighborhoods. The fundamental properties of these classes, as well as their relationships with $\mathit{j}$-neighborhoods and ordered equivalence classes under various binary relations, are then examined. Building on these classes, ordered $\mathit{j}$-approximation spaces are introduced as a natural generalization of both ordered approximation spaces and Pawlak approximation spaces. The findings show that the proposed approximations yield more capable results in reducing boundary regions compared to some existing approaches. Moreover, generalized nano topologies are constructed from these new structures, and their applicability is demonstrated through a real-world example.