Soft Symmetric Difference Complement-Intersection Product of Groups

Year 2025, Volume: 8 Issue: 2, 71 - 82

Zeynep Ay , Aslıhan Sezgin

https://doi.org/10.70030/sjmakeu.1733234

Abstract

Soft set theory, as a mathematically rigorous and algebraically expressive formalism, offers a powerful framework for modeling uncertainty, vagueness, and parameter-driven variability. Within this landscape, the present study introduces the soft symmetric difference complement-intersection product, a novel binary operation defined over soft sets whose parameter domains are endowed with a group-theoretic structure. Developed within a strict axiomatic foundation, the operation is proven to satisfy fundamental algebraic properties—such as closure, associativity, commutativity, and idempotency—while maintaining consistency with generalized notions of soft equality and subsethood. Its behavior is thoroughly analyzed with respect to identity and absorbing elements, as well as interactions with null and absolute soft sets, all within the constraints of group-parameterized domains. The findings confirm that the proposed operation forms a coherent and structurally robust algebraic system, thereby enriching the algebraic architecture of soft set theory. Furthermore, this work provides a foundational step toward the formulation of a generalized soft group theory, in which soft sets indexed by group-based parameters emulate classical group behaviors through abstract soft operations. The operation’s full integrability within soft inclusion hierarchies and its alignment with generalized soft equalities highlight its theoretical depth and broaden its potential applications in formal decision-making and algebraic modeling under uncertainty.

Keywords

Soft sets , Soft subsets , Soft equalities , Soft symmetric difference complement-intersection

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