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Published:
May 2, 2026DOI:
10.4314/mejs.v18i1.1Keywords:
Article Details
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Main Article Content
Soft Symmetric Difference-Intersection Product of Groups
Aslıhan Sezgin
Zeynep Ay
Abstract
Soft set theory has emerged as a comprehensive mathematical apparatus for the systematic representation and analysis of uncertainty, particularly in the environments governed by parametric variability. At the heart of this theoretical construct lie soft set operations and products, which collectively furnish powerful tools for the formulation and resolution of complex problems characterized by parameter-driven indeterminacy. The present work initiates with a meticulous and formal investigation of the symmetric difference operation of soft sets. It is rigorously established that the collection of the soft sets with a fixed parameter set, under this binary operation, forms an abelian group, thereby endowing the structure with foundational algebraic coherence. Subsequently, we introduce an original product, termed the soft symmetric difference–intersection product, defined on soft sets whose associated parameter sets are endowed with a group structure. A thorough axiomatic and structural investigation of this product is undertaken, with particular emphasis on its compatibility with existing notions of soft subsethood and equality. It is further demonstrated that the algebraic structure comprising the set of soft sets with a fixed parameter set, together with the symmetric difference of soft sets and the proposed product, satisfies the axiomatic requirements of both a ring and a hemiring. The theoretical implications of these findings are twofold: on the one hand, they significantly advance the algebraic underpinnings of soft set theory; on the other, they open new vistas for the formulation of a nascent soft group theory, potentially parallel to its classical counterpart. Given that the formal development of soft algebraic frameworks is intrinsically contingent upon rigorously defined operations and product constructions, the present study constitutes a substantial and foundational contribution to the algebraic maturation of soft set theory.
