Fuzzy MultiDirected Sets, Neutrosophic MultiDirected Sets, Plithogenic MultiDirected Sets, Soft MultiDirected Sets, and Hypersoft MultiDirected Sets

Abstract

A fuzzy set generalizes classical sets by assigning each element a membership value in the interval [0, 1], thereby capturing partial or uncertain membership. A Neutrosophic Set extends fuzzy sets by introducing three independent membership degrees: truth, indeterminacy, and falsity, each ranging within [0, 1]. A Plithogenic Set further generalizes classical and fuzzy sets by incorporating attributes, their possible values, and a measure of contradiction. Other notable frameworks for handling uncertainty include the Soft Set and Hypersoft Set, which provide flexible parameterized approaches. Additionally, concepts such as the Directed Set and MultiDirected Set, which introduce directionality to sets, are also well-established. This paper explores and analyzes the structures of Fuzzy Directed Sets, Neutrosophic Directed Sets, Plithogenic Directed Sets, Fuzzy MultiDirected Sets, Neutrosophic MultiDirected Sets, Plithogenic MultiDirected Sets, Soft Directed Sets, Hypersoft Directed Sets, Soft MultiDirected Sets, and Hypersoft MultiDirected Sets.