A Comprehensive Study on Restricted and Extended Intersection Operations of Soft Sets

Soft set theory has gained prominence as a revolutionary approach for handling uncertainty-related problems and modeling uncertainty since it was proposed by Molodtsov. The concept of soft set operations, which is the major notion for the theory, has served as the foundation for theoretical and practical advances in the theory, therefore deriving the algebraic properties of the soft set operations and studying the algebraic structure of soft sets associated with soft set operations have attracted the researchers’ interest continuously. In the theory of soft set, many soft intersection operations have been defined up to now among which there are some differences, and some of which are no longer preferred for use as they are essentially not useful and functional. Although the definition of restricted intersection is widely accepted in the literature and used in the studies, it is still incomplete with its current form suffering from certain cases where the parameter sets of the soft sets may be disjoint is ignored, thus all the circumstances in the theorems are not considered in the related proofs causing to the incorrectness or deficiency in the studies where this operation is used or its properties are investigated. In this regard, in the existing literature, there is a critical lack of comprehensive study on the correct defined restricted intersection operation together with extended intersection including their correct properties and distributions and the correct algebraic structures assoiciated with these soft set operations. In this study, we primarly intend to fill this crucial gap by first correcting the deficiencies in the presentation of the definition of restiricted intersection and revising it. Moreover, in many papers related to these operation, several theorems were presented without their proofs, or there were some incorrect parts in the proofs. In this study, all the proofs based on the function-equality are regularly provided and besides, the relationships between the concept of soft subset and restricted and extended intersection operations are presented for the first time with their detailed proofs. Furhermore, we obtain many new properties of these operations as analogy and counterpart of intersection operation in classical set theory. Moreover, the operations’ full properties and distributions over other soft set operations are throughly investigated to determine the correct algebraic structures the operations form individually and in combination with other soft set operations both in the set of soft sets over the universe and with a fixed parameter set. We demonstrate that the restricted/extended intersection operations, when combined with other kinds of soft set operations, form several significant algebraic structures, such as monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, Kleene Algebra, Stone algebra and MV-algebra but with deteailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted intersection and extended intersection 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 filling the significant gap in the literature, acting as a guide for the beginners of this popular theory, and besides shedding light on the future studies on soft sets.