Soft Symmetric Difference Complement-Theta Product of Groups

Abstract

In complex decision-making and information systems, uncertainty ambiguity, and parameter-based variability can be effectively modeled using soft set theory, a mathematically coherent and algebraically expressive framework.  We present a new binary operation on soft sets whose parameter sets are group-theoretic structures: the soft symmetric difference complement–theta product. The operation's algebraic coherence within the larger soft set framework is ensured by its rigorous axiomatic foundation, which maintains full compatibility with expanded concepts of soft equality and soft subset-hood. Key structural properties—closure, associativity, commutativity, and idempotency—as well as their connections with identity, absorbing, null, and absolute soft sets are established by a thorough algebraic study. The outcomes verify that all algebraic constraints imposed by group-based parameterization are satisfied by the operation. Beyond its fundamental significance, the suggested operation enhances soft set theory's structural landscape and offers a formal link to generalized soft group theory, in which soft sets indexed by group elements mimic classical algebraic behavior. Thus, this contribution is a significant step toward applying soft set theory to fields that require rigorous tools for multi-criteria decision processes, abstract algebraic representation, and uncertainty modeling.