SOFT SYMMETRIC DIFFERENCE COMPLEMENT WITH SOFT INTERSECTION-SYMMETRIC DIFFERENCE COMPLEMENT PRODUCT OF GROUPS

Year 2025, Volume: 11 Issue: 2, 194 - 211, 30.12.2025

Aslıhan Sezgin , İbrahim Durak , Zeynep Ay

https://doi.org/10.51477/mejs.1751497

Abstract

Soft set theory provides a mathematically rigorous and algebraically expressive formalism for representing systems marked by uncertainty and parameter-dependent variability. Within this formal context, the present study introduces the soft symmetric difference complement of soft sets and soft intersection–symmetric difference complement product of groups, a novel product defined on soft sets whose parameter domains are structured by inherent group-theoretic properties. A comprehensive algebraic analysis is conducted to establish the operation’s and the proposed product’s fundamental structural invariants, including closure, associativity, commutativity, identity, inverse and absorbing element and idempotency. Their interactions with null and absolute are also rigorously characterized, and their properties are examined in relation to the algebraic constraints dictated by group-parameterized domains. It is rigorously demonstrated that the collection of all soft sets defined over a fixed parameter set forms an abelian group under the symmetric difference complement of soft sets, thereby imparting foundational algebraic structure to the soft set framework. Furthermore, the soft symmetric difference complement together with the union operation of soft sets forms a commutative hemiring with identity in the collections of soft sets over the universe likewise the symmetric difference complement together with the union operations of sets forms a commutative hemiring with identity in the power set of the universe. This result not only strengthens the internal consistency of soft set theory but also establishes a basis for extending classical algebraic concepts to soft environments. Moreover, by confirming the associativity, commutativity, identity, and invertibility properties within this structure, the study paves the way for constructing more complex algebraic systems—such as rings and modules—built upon soft set operations. This study not only advances the formal algebraic apparatus of soft set theory but also provides a principled framework for addressing uncertainty in abstract algebraic modeling, logical systems, and multi-criteria decision-making environments.

Keywords

Soft sets , Soft subsets , Soft equalities , Soft symmetric difference complement operations , Soft intersection-symmetric difference complement product , Hemiring

Ethical Statement

The authors declare that this document does not require ethics committee approval or any special permission. Our study does not cause any harm to the environment and does not involve the use of animal or human subjects.

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