Since its introduction by Molodtsov, soft set theory has developed in prominence as an innovative method for dealing with uncertainty-related problems and modeling uncertainty. Soft set operations, the theory's main concept, have served as the foundation for theoretical and practical advances in the theory; thus, deriving the algebraic properties of soft set operations and studying the algebraic structure of soft sets associated with soft set operations has piqued researchers' interest continuously. Many soft union operations have been proposed in soft set theory, but there are some differences. Even though the definition of restricted union is widely acknowledged in the literature and applied in many studies, it is still lacking in its current form owing to the fact that a specific case where the soft sets' parameter sets may be disjoint is ignored in the definition. As a result, all the cases in the theorems are not taken into account, leading to errors or deficiencies in the studies that use this operation or investigate this operation’s properties. Regarding this, a thorough examination of the properly defined restricted union operation and extended union operation, together with their appropriate distributions and properties, as well as the appropriate algebraic structures connected to these soft set operations, is conspicuously lacking in the body of current literature. This study is primarily intended to fill this critical and significant gap by first fixing the presentational flaws of the restricted union definition and updating it. Furthermore, in many works on these operations, numerous theorems were offered without their proofs, or the proofs had some incorrect parts. In this paper, all the proofs based on function equality are supplied regularly, and the relations between the concept of soft subset and restricted and extended union operations are obtained for the first time with their extensive proofs. Furthermore, we explore numerous novel properties of these operations as analogies and counterparts of the union operation in classical set theory. We show that when restricted/extended union operations are combined with other types of soft set operations, several significant algebraic structures are formed, including monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, and MV-algebra with detailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted union and extended union in the literature to date, as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by bridging a significant gap in the literature, serving as a guide for newcomers to this popular theory, and providing insight for further future research.
Authors: Aslıhan Sezgin, Hakan Kökçü, Nenad Stojanović, Murat Luzum
DOI: https://doi.org/10.4314/ijest.v17i1.1
Publish Year: 2025
