Soft Intersection-star Product of Groups

Soft set theory provides a mathematically rigorous and algebraically expressive framework for modeling systems characterized by epistemic uncertainty, vagueness, and parameter-dependent variability—phenomena central to decision theory, engineering, economics, and information science. Expanding on this foundation, the present study introduces and examines a novel binary operation, the soft intersection–star product, defined over soft sets with parameter domains possessing intrinsic group-theoretic structures. Developed within a formally consistent, axiomatic framework, this operation aligns with generalized concepts of soft subsethood and soft equality. A comprehensive algebraic analysis is on the operation’s core properties—closure, associativity, commutativity, and idempotency. The presence or absence of identity, inverse, and absorbing elements, and the soft product’s behavior concerning the null and absolute soft sets, are precisely delineated. To contextualize the operation, a comparative analysis with prior binary soft products is conducted, elucidating its expressive capacity and structural coherence within the layered hierarchy of soft subset classifications. The findings demonstrate that the soft in-tersection–star product satisfies all axiomatic requirements imposed by group-parameterized domains, thereby inducing a robust and internally consistent algebraic structure on the space of soft sets. Two key contributions emerge: first, the operation substantially extends the algebraic toolkit of soft set theory within a rigorous opera-tional framework; second, it lays the foundation for a generalized soft group theory, wherein soft sets indexed by group-structured parameters mimic classical group behavior through abstractly defined soft operations. Beyond its theoretical value, the proposed framework offers a principled basis for soft computational modeling grounded in abstract algebra. Such models are highly applicable to multi-criteria decision analysis, algebraic classification, and uncertainty-sensitive data analytics. Hence, this study not only strengthens the theoretical foundations of soft algebra but also reinforces its relevance to both mathematical research and practical computation.