Soft Symmetric Difference Complement-Difference Product of Groups - Durak, İbrahim and Sezgin, Aslıhan (Soft Computing Fusion with Applications, 2025)
Soft set theory, known for its mathematical rigor and algebraic expressiveness, provides a robust framework for addressing uncertainty, vagueness, and variability driven by parameters. This study presents a new binary operation called the soft symmetric difference complement-difference product, which is defined over soft sets with parameter domains that have a group-theoretic structure. Built on a solid axiomatic basis, this operation is shown to fulfill essential algebraic properties, including closure, associativity, commutativity, and idempotency, while aligning with broader concepts of soft equality and subset relationships. The study thoroughly examines the operation's characteristics regarding identity and absorbing elements, as well as its interactions with null and absolute soft sets, all within the context of group-parameterized domains. The results indicate that this operation creates a coherent and structurally sound algebraic system, enhancing the algebraic framework of soft set theory. Additionally, this research lays the groundwork for developing a generalized soft group theory, where soft sets indexed by group-based parameters mimic classical group behaviors through abstract soft operations. The operation's complete integration within soft inclusion hierarchies and its compatibility with generalized soft equalities underscore its theoretical significance and expand its potential uses in formal decision-making and algebraic modeling under uncertainty.