Soft Union-Difference Product of Groups

Soft set theory constitutes a highly flexible and mathematically rigorous framework for modeling and analyzing real-world phenomena characterized by uncertainty, ambiguity, and parameter-dependent variability—features that frequently arise in disciplines such as decision sciences, engineering, economics, and information systems. Central to this theoretical apparatus are the fundamental operations and product constructions on soft sets, which collectively give rise to a rich and expressive algebraic infrastructure capable of accommodating complex parametric interdependencies. In this study, we introduce a novel product, termed the soft union–difference product, specifically defined for soft sets whose parameter sets possess a group structure. A thorough axiomatic and structural analysis of this is conducted, with special attention to its algebraic compatibility with generalized notions of soft subsethood and soft equality. Through this analysis, we uncover the product’s intrinsic structural properties and demonstrate its capacity to preserve essential algebraic features within group-parameterized soft set systems. Furthermore, we conduct a comprehensive algebraic investigation of the soft union–difference product, examining its closure, associativity, idempotency, commutativity, absorbing property, and distributivity, as well as its interaction with other established soft products defined on groups and null soft sets. These investigations reveal two pivotal theoretical implications: first, they reinforce the internal algebraic coherence of soft set theory by situating the newly defined product within a formally consistent operational framework; second, they lay a conceptual foundation for the emergence of a soft group theory that structurally parallels classical group-theoretic constructions. Given that the advancement of soft algebraic systems is inherently predicated on rigorously defined operations and systematically articulated product frameworks, the present study makes a substantial contribution to the formal algebraic refinement and theoretical evolution of soft set theory. Beyond their theoretical merit, the proposed constructions also offer concrete methodological tools for the development of group-based soft computational models, with potential applications in multi-criteria decision-making, uncertainty-aware classification systems, and data-driven analysis under parameter uncertainty.