Soft union-star product of groups

Soft set theory constitutes a logically rigorous and algebraically expressive formalism for representing systems permeated by ambiguity, epistemic uncertainty, and parameter-dependent variability. In this context, the present study introduces the soft union–star product, a novel binary operation defined on soft sets whose parameter do-mains are endowed with an intrinsic group-theoretic structure. Formulated within a strictly axiomatic framework, the operation is proven to exhibit full compatibility with generalized formulations of soft subsethood and soft equality. A comprehensive algebraic analysis is conducted to establish its core structural invariants, including closure, associativity, commutativity, and idempotency. Moreover, the operation’s behavior is rigorously characterized in relation to the identity and absorbing elements, as well as its interaction with the null and absolute soft sets. The findings confirm that the soft union–star product satisfies all algebraic conditions imposed by group-parameterized domains, thereby generating a robust and internally coherent algebraic structure over the universe of soft sets. Beyond its foundational significance, the operation meaningfully enriches the operational landscape of soft set theory and provides a formal platform for advancing a generalized soft group theory. Its structural compatibility with key relational constructs—particularly generalized soft equalities and inclusion hierarchies—underscores its potential utility in diverse application domains, including algebraic abstraction, uncertainty-sensitive classification, and multi-criteria decision-making. As such, this work contributes not only a substantive theoretical advancement but also a mathematically principled pathway for practical deployment in uncertainty-aware systems.