Soft symmetric difference complement-union product of groups

Abstract

Soft set theory, recognized for its mathematical precision and algebraic capabilities, offers a strong framework for tackling uncertainty, ambiguity, and variability influenced by parameters. This research introduces a novel binary operation known as the soft symmetric difference complement-union product, which is defined for soft sets with parameter domains that exhibit a group-theoretic structure. Based on a solid axiomatic foundation, this operation is demonstrated to satisfy key algebraic properties such as closure, associativity, commutativity, and idempotency, while also being consistent with broader notions of soft equality and subset relationships.  It is obtained that the proposed product is a noncommutative semigroup in the collections of soft sets with a fixed parameter set.The study provides an in-depth analysis of the operation's features concerning identity and absorbing elements, as well as its interactions with null and absolute soft sets, all within the framework of group-parameterized domains. The findings suggest that this operation establishes a coherent and structurally robust algebraic system, thereby enhancing the algebraic framework of soft set theory. Furthermore, this research sets the stage for the development of a generalized soft group theory, where soft sets indexed by group-based parameters emulate classical group behaviors through abstract soft operations. The operation's full integration within soft inclusion hierarchies and its compatibility with generalized soft equalities highlight its theoretical importance and broaden its potential applications in formal decision-making and algebraic modeling under uncertainty.