Soft symmetric difference complement-gamma product of groups

Authors : İbrahim Durak, Aslıhan Sezgin

Pages : 74-85

Doi:10.55117/bufbd.1751678

View : 25 | Download : 41

Publication Date : 2025-12-31

Article Type : Research Paper

Abstract :As a mathematically consistent and algebraically potent formal system, soft set theory is well-suited for capturing the inherent uncertainty, vagueness, and parameter-based variability in complex decision-making and information systems. Within this theoretical framework, the present study introduces the soft symmetric difference complement–gamma product, a novel binary operation defined on soft sets whose parameter spaces are intrinsically structured by group-theoretic properties. Formulated through a rigorous axiomatic foundation, the operation is shown to maintain full compatibility with generalized soft subsethood and soft equality, thereby ensuring its structural integration within the broader algebraic fabric of soft set theory. A comprehensive algebraic examination is conducted to identify and verify the operation’s key structural properties, including closure, associativity, commutativity, and idempotency. The operation’s interactions with identity and absorbing elements, as well as with the null and absolute soft sets, are rigorously analyzed within the constraints imposed by group-parameterized domains. The findings confirm that the soft symmetric difference complement–gamma product satisfies all required algebraic conditions dictated by group-theoretic parameter structures, resulting in a coherent and robust algebraic framework over the universe of soft sets. In addition to its foundational value, the operation substantially enriches the structural landscape of soft set theory and offers a formal pathway toward the construction of a generalized soft group theory, where soft sets indexed by group-based parameters emulate classical algebraic operations. Its consistency with generalized notions of soft equality and its seamless integration into layered soft inclusion hierarchies further emphasize its algebraic depth and applicability. Accordingly, this work not only delivers a significant algebraic innovation but also provides a principled platform for extending soft set theory to fields that demand formal mechanisms for handling uncertainty, abstract algebraic modeling, and multi-criteria decision-making frameworks.
Keywords : Esnek kümeler, Esnek alt kümeler, Esnek eşitlikler, esnek simetrik fark tümleyen-gama çarpımı

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