Generalized Ordered Rough Set Models via Minimal and Maximal Neighborhoods Induced by Binary Relations

Classical Pawlak rough set theory is fundamentally based on equivalence relations, which limits its ability to represent uncertainty arising from non-equivalent and ordered information. This paper develops a generalized ordered rough set framework in which binary relations are combined with partial order structures to construct minimal and maximal neighborhood systems. Based on these neighborhoods, generalized increasing and decreasing approximation operators are introduced and systematically related to the corresponding right-neighborhood model. The proposed framework establishes fundamental inclusion, boundary, and accuracy relationships between the minimal and maximal models, showing in particular that minimal neighborhoods generally yield tighter approximations, whereas maximal neighborhoods produce wider uncertainty regions. Six associated nano-topological structures, together with their basis representations, are constructed for the right, minimal, and maximal neighborhood systems, providing a unified topological setting for the analysis of ordered uncertainty. The theoretical developments are further connected with several established rough set models, demonstrating the generality of the proposed constructions. A topological attribute-reduction procedure is then developed to identify dispensable and indispensable attributes through changes in the induced structural patterns. In addition, the proposed m-Nano Flou measure is employed as a complementary structural-sensitivity criterion for quantitatively characterizing the effects of attribute removal. A COVID-19 medical information system adopted from a published study is used to illustrate the applicability of the framework to ordered medical information. The obtained rankings are interpreted exclusively in terms of structural influence within the proposed approximation framework and do not represent clinical importance or diagnostic performance. The proposed approach therefore provides a flexible mathematical framework for modeling and analyzing uncertainty in ordered information systems beyond the restrictive equivalence-relation setting of classical rough sets.