Soft Symmetric Difference-lambda Product of Groups

Authors : Zeynep Ay, Aslıhan Sezgin

Pages : 155-171

Doi:10.57244/dfbd.1729245

View : 22 | Download : 34

Publication Date : 2025-12-30

Article Type : Research Paper

Abstract :Soft set theory offers a mathematically robust and algebraically expressive formalism for modeling systems characterized by epistemic uncertainty, vagueness, and parameter-dependent variability—core features of decision theory, engineering, economics, and information sciences. Building upon this foundation, the present study introduces and investigates a novel binary operation, referred to as the soft symmetric difference–lambda product, defined over soft sets whose parameter domains are endowed with group-theoretic structure. This operation is rigorously formalized within an axiomatic framework that ensures compatibility with generalized soft subsethood and soft equality relations, thereby preserving the internal consistency and structural coherence of the induced algebraic system. A comprehensive algebraic analysis is carried out to establish the fundamental properties of the proposed operation, including closure, associativity, commutativity, idempotency, and distributivity over other soft set operations. The behavior of the product with respect to identity and absorbing elements, as well as its interactions with the null and absolute soft sets, is explicitly characterized. To situate the proposed operation within the broader algebraic landscape of soft set theory, a comparative study is conducted with existing binary soft products, highlighting its expressive strength, structural alignment, and integrability within established soft subset hierarchies. The results demonstrate that the soft symmetric difference–lambda product satisfies the axiomatic requirements imposed by group-parameterized domains and induces a well-behaved, formally consistent algebraic structure over the space of soft sets. Two principal contributions emerge from this investigation. First, the introduction of this product enriches the algebraic toolkit of soft set theory by embedding it in a rigorous, operation-preserving environment. Second, it provides a foundational platform for the development of a generalized soft group theory, wherein soft sets indexed by group-structured parameters simulate classical group-theoretic behavior through soft operations. Beyond its theoretical significance, the algebraic framework proposed herein offers a principled basis for the construction of soft computational models grounded in abstract algebra, with potential applications in multi-criteria decision-making, algebraic classification systems, and uncertainty-aware data analysis. Accordingly, the formal structure developed in this study not only advances the theoretical generalization of soft algebra but also reinforces its practical utility across both abstract and applied domains.
Keywords : Esnek kümeler, Esnek alt kümeler, Esnek eşitlikler, Esnek simetrik fark-lamda çarpımı

ORIGINAL ARTICLE URL