Soft union-plus product of groups

Abstract

A rigorous and expressive algebraic framework for modeling systems with uncertainty, ambiguity, and parameter-dependent variability is provided by soft set theory. In this paper, we present a new binary operation on soft sets whose parameter domains have group-theoretic structure: the soft union–plus product. The operation is completely compatible with generalized concepts of soft equality and soft subsethood when specified formally inside an axiomatic framework. Key structural aspects such as closure, associativity, commutativity, idempotency, and distributivity are investigated in detail algebraically, includinng its behavior with respect to identity, absorbing, null, and absolute soft sets. The outcomes demonstrate that the operation creates a strong and cohesive algebraic system on the universe of soft sets while adhering to all algebraic limitations imposed by group-indexed domains. In addition to its theoretical importance, the operation provides a strong basis for a generalized soft group theory and reinforces the underlying algebraic architecture of soft set theory. Furthermore, it has significant potential for both abstract theoretical advancement and real-world applications due to its formal consistency with soft subset and equality relations, which improves its usefulness in domains like categorization, decision-making, and uncertainty-aware modeling.