Precision in Soft Rough Sets: Covering-Based Methods Enhanced by Ideals

Abstract

This paper presents a generalized framework for covering-based soft rough sets by incorporating ideals from topological spaces to improve the accuracy and precision of approximations. Building on the foundational concepts of soft sets and soft rough sets, we introduce refined definitions for soft lower and upper approximations using ideals and establish their mathematical properties through a series of theorems. We show that while some classical rough set properties are preserved under this generalization, others may not hold, highlighting the nuanced behaviour of ideal-based approximations. The approximation process is further enhanced by defining more precise notions of boundary, positive region, and negative region, along with an improved accuracy measure that better captures the effectiveness of approximation in soft covering spaces. It is also shown that the ideal-based lower approximation satisfies Kuratowski’s closure axioms, inducing a topology on the universe. Through illustrative examples and an applied decision-making scenario, we demonstrate that the proposed ideal-based approach provides higher accuracy compared to classical methods, particularly in contexts involving overlapping or imprecise data. This work offers a topologically grounded enhancement to soft rough set theory and contributes practical tools for more reliable data analysis and decision-making in uncertain environments.