Matrix Science Mathematic (MSMK)

Soft set theory constitutes a mathematically robust and structurally versatile formalism for modeling real-world systems characterized by epistemic uncertainty, vagueness, and parameter-contingent variability—ubiquitous features across decision theory, engineering, economics, and information sciences. At the core of this framework lies a spectrum of algebraic operations and binary product constructions that endow the soft set universe with a rich internal structure, capable of encapsulating intricate interdependencies among parameters. In this context, we introduce and investigate a novel product of soft sets, termed the soft intersection–symmetric difference product, formulated specifically for soft sets whose parameter domains are structured as groups. This product is rigorously defined and analyzed within an axiomatic framework that ensures compatibility with generalized soft subsethood and equality relations. The structural analysis of the soft intersection–symmetric difference product includes the examination of essential algebraic properties—such as closure, associativity, commutativity, and idempotency. In addition, the interplay between this product and pre-existing soft products, is explored to regarding the subsets. Theoretical investigations reveal that the operation not only respects the algebraic architecture of the underlying group-parameterized domain but also induces a cohesive and well-behaved algebraic system on the collection of soft sets. This analytical framework yields two central algebraic insights: (i) the internal algebraic cohesion of soft set theory is significantly enhanced by embedding the newly defined product into a logically sound and operation-preserving environment; and (ii) the product itself possesses the formal potential to serve as a foundational construct for a generalized soft group theory, wherein soft sets over group-parameter spaces mimic the axiomatic behavior of classical group-theoretic constructs through suitably defined soft operations. Given that the maturation of soft algebraic systems is contingent upon the rigorous formulation of operations satisfying structurally meaningful axioms, the contributions of this study represent a notable advancement in the algebraic consolidation of soft set theory. Beyond theoretical enrichment, the proposed operation offers tangible utility in the construction of abstract algebra-based soft computational models, with applications spanning multi-criteria decision-making, algebraically-driven classification mechanisms, and uncertainty-aware data analysis governed by group-parametrized semantic domains. Thus, the framework established herein not only extends the theoretical boundaries of soft algebra but also fortifies its role as a foundational tool in both pure and applied mathematical discourse