Soft Intersection-plus Product of Groups

Authors : Zeynep Ay, Aslıhan Sezgin

Pages : 1-25

View : 24 | Download : 34

Publication Date : 2025-10-01

Article Type : Research Paper

Abstract :Soft set theory constitutes a mathematically rigorous and algebraically expansive framework for representing and analyzing systems permeated by epistemic uncertainty, vagueness, and parameter-contingent variability—hallmarks of foundational problems in decision sciences, engineering, economics, and information theory. Central to this formalism is a comprehensive repertoire of algebraic operations and binary product constructions, which together endow the universe of soft sets with a nuanced internal algebraic topology capable of encoding complex interdependencies among parameters with high structural fidelity. Within this algebraic landscape, we formally introduce and investigate a novel binary operation—termed the soft intersection–plus product—defined on soft sets whose parameter sets are structured as groups. The operation is rigorously developed within an axiomatic framework that guarantees compatibility with generalized notions of soft subsethood and soft equality, thereby maintaining the algebraic integrity of the induced structure. A meticulous algebraic analysis is carried out to establish key structural attributes of the operation, including closure, associativity, commutativity, idempotency, distributivity over other sot set operataions, as well as its behavior with respect to identity and absorbing elements, and its interaction with the null and absolute soft sets. Moreover, the proposed operation is systematically compared with pre-existing soft binary operations within a stratified taxonomy of soft subset classifications, yielding deeper theoretical insights into their comparative expressive strength and algebraic integrability. Our findings reveal that the soft intersection–plus product not only conforms to the algebraic constraints imposed by group-parameterized domains, but also induces a well-behaved and internally coherent algebraic system over the soft set space. Two principal contributions emerge from this investigation: (i) the integration of the proposed product enhances the internal algebraic cohesion of soft set theory by embedding it within a formally consistent, axiom-preserving framework; and (ii) it serves as a foundational component for the development of a generalized soft group theory, wherein soft sets indexed by group-structured parameter domains emulate classical group-theoretic behavior via newly defined soft operations. By addressing the critical need for algebraic constructions grounded in both semantic relevance and structural rigor, this study represents a substantial advance in the algebraic consolidation and theoretical generalization of soft set theory. Beyond its abstract significance, the proposed operation offers a robust mathematical basis for the design of soft computational models governed by algebraic principles, with prospective applications in multi-criteria decision-making, algebraic classification frameworks, and uncertainty-aware data analysis across group-parameterized semantic environments. As such, the formal apparatus developed herein not only expands the theoretical frontier of soft algebra but also affirms its relevance in both pure mathematics and applied analytical disciplines.
Keywords : Esnek kümeler, Esnek alt kümeler, Esnek eşitlikler, Esnek kesişişm-artı çarpımı

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