Some New Classifications of Soft Subsets and Soft Equalities with Soft Symmetric Difference-Difference Product of Groups

Soft sets provide a comprehensive mathematical framework for tackling uncertainty. Soft set operations and products are fundamental to soft set theory, offering innovative solutions to problems that involve parametric data. In this study, we first adapted the soft L-subsets/equality concept and soft J-subsets/equality for the revised soft set concept. Additionally, we defined some new types of soft subsets and equalities, called soft S-subsets/equality and soft A-subsets, along with their specific examples to clarify these concepts. We investigated the connections among these new concepts. This paper presents an innovative product for soft sets whose parameter sets are groups, called the "soft symmetric difference-difference-product". We thoroughly analyzed its fundamental algebraic properties, considering various soft subsets and equality relations to inspire future research. This may lead to a new soft group theory arising from this concept. Since soft algebraic structure theories are grounded in soft set operations and products, this study contributes significantly to the literature on soft sets.