Rough Approximations of Soft Ideals in Lattices Under Soft Relation: A Multi-Criteria Group Decision-Making Perspective

Rough sets (RSt) and soft sets (SSt) are two exceptional mathematical frameworks for handling vague and imprecise data in real-world applications. This work introduces a novel method for addressing the roughness of soft substructures in lattices, specifically related to foresets (Fsets) and aftersets (Asets), by utilizing soft binary relations (SBREs). This type of roughness is more general than the roughness of SSt in lattices performed by a congruence relation and set-valued mappings. We investigate the roughness of soft sublattice under SBRE defined on the lattice and studied some relevant structural properties. Moreover, the idea is extended to rough soft ideals, rough soft prime ideals, rough soft filters and rough soft prime filters under SBREs. We explore some new soft rough operations. Additionally, we observe that soft compatible and soft complete relations are required while evaluating soft rough substructures of lattices. Further, we establish a relationship between the approximations of soft substructures of lattices and those of their homomorphic images, utilizing lattice homomorphism and SBRE. Lastly, we present a method based on rough soft sets defined on SBREs for the multicriteria group decision-making problem, along with a real-world example.