SOFT SYMMETRIC DIFFERENCE-PLUS PRODUCT OF GROUPS

Soft set theory constitutes a mathematically rigorous and algebraically versatile framework for modeling systems characterized by epistemic uncertainty, vagueness, and parameter-dependent variability. Building upon this foundational structure, the present study introduces and thoroughly investigates a novel binary operation, termed the soft symmetric difference-plus product, defined on soft sets whose parameter set is a group. This operation is axiomatized within a logically coherent and formally consistent framework, ensuring full compatibility with generalized notions of soft subsethood and soft equality. A comprehensive algebraic analysis is undertaken to establish the fundamental properties of the operation, including closure, associativity, commutativity, and idempotency. Moreover, the existence or absence of the identity and the absorbing elements of the product, along with its characteristics relative to the null and absolute soft sets, are explicitly characterized. This inquiry elucidates the relative expressive capacity, algebraic coherence, and structural integrability of the soft symmetric difference-plus product within the layered hierarchies of soft subset classifications. The results demonstrate that the operation satisfies all necessary axiomatic requirements dictated by group-parameterized domains, thereby inducing a robust and internally consistent algebraic structure on the universe of soft sets. Two principal contributions emerge: first, the introduction of the soft symmetric difference-plus product substantially enriches the operational repertoire of soft set theory by embedding it within a rigorously defined, operation-preserving algebraic framework; second, it lays a conceptual foundation for the advancement of a generalized soft group theory, wherein soft sets indexed by group-structured parameter domains emulate classical group behaviors through abstractly formulated soft operations. Beyond its theoretical significance, the framework developed herein provides a mathematically principled basis for constructing soft computational models grounded in abstract algebra. Such models hold considerable promise for applications in multi-criteria decision-making, algebraic classification, and uncertainty-aware data analysis.