Soft Union-lambda Product of Groups

Soft set theory provides a logically sound and algebraically rich framework for modeling systems characterized by ambiguity, epistemic uncertainty, and parameter-dependent variability. This study introduces the soft union–lambda product, a novel binary operation defined over soft sets whose parameter spaces are endowed with an intrinsic group-theoretic structure. Developed within a rigorously formulated axiomatic framework, the opera-tion is shown to be fully compatible with generalized notions of soft subsethood and soft equality. A thorough algebraic investigation is undertaken to establish the fundamental structural properties of the opera-tion—including closure, associativity, commutativity, idempotency, and its distributivity over other soft set operations—alongside a precise characterization of its behavior with respect to identity, absorbing, null, and absolute soft sets. The results demonstrate that the soft union–lambda product adheres to all algebraic con-straints dictated by group-parameterized domains, thereby inducing a consistent and internally cohesive alge-braic structure on the universe of soft sets. Beyond its foundational contributions, the proposed operation sig-nificantly expands the formal toolkit of soft set theory and sets the stage for the development of a generalized soft group theory. Furthermore, its formal alignment with key relational structures such as soft equality and soft inclusion highlights its potential applicability across a diverse array of analytical contexts, including abstract algebraic modeling, uncertainty-aware classification, and multi-criteria decision analysis. Accordingly, the findings of this study offer both profound theoretical advancements and concrete pathways for practical im-plementation.