A Dynamic Survey of Soft Set Theory and Its Extensions

Soft set theory provides a direct framework for parameterized decision modeling by assigning to each attribute (parameter) a subset of a given universe, thereby representing uncertainty in a structured way [1, 2]. Over the past decades, the theory has expanded into numerous variants-including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets-and has been connected to diverse areas such as topology and matroid theory. In this book, we present a survey-style overview of soft sets and their major extensions, highlighting core definitions, representative constructions, and key directions of current development.

Superhypergraph Neural Networks and Plithogenic Graph Neural Networks: Theoretical Foundations

Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GNNs), a well-established framework, have recently been extended to Hypergraph Neural Networks (HGNNs), with their properties and applications being actively studied. The Plithogenic Graph framework enhances graph representations by integrating multi-valued attributes, as well as membership and contradiction functions, enabling the detailed modeling of complex relationships. In the context of handling uncertainty, concepts such as Fuzzy Graphs and Neutrosophic Graphs have gained prominence. It is well established that Plithogenic Graphs serve as a generalization of both Fuzzy Graphs and Neutrosophic Graphs. Furthermore, the Fuzzy Graph Neural Network has been proposed and is an active area of research. This paper establishes the theoretical foundation for the development of SuperHyperGraph Neural Networks (SHGNNs) and Plithogenic Graph Neural Networks, expanding the applicability of neural networks to these advanced graph structures. While mathematical generalizations and proofs are presented, future computational experiments are anticipated.

Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond

To better handle real-world uncertainty, concepts such as fuzzy sets, neutrosophic sets, rough sets, and soft sets have been introduced. For example, neutrosophic sets, which simultaneously represent truth, indeterminacy, and falsehood, have proven to be valuable tools for modeling uncertainty in complex systems. These set concepts are increasingly studied in graphized forms, and generalized graph concepts now encompass well-known structures such as hypergraphs and superhypergraphs. Furthermore, hyperconcepts and superhyperconcepts are being actively researched in areas beyond graph theory. Combinatorics, uncertain sets (including fuzzy sets, neutrosophic sets, rough sets, soft sets, and plithogenic sets), uncertain graphs, and hyper and superhyper concepts are active areas of research with significant mathematical and practical implications. Recognizing their importance, this paper explores new graph and set concepts, as well as hyper and superhyper concepts, as detailed in the "Results" section of "The Structure of the Paper." Additionally, this work aims to consolidate recent findings, providing a survey-like resource to inform and engage readers. For instance, we extend several graph concepts by introducing Neutrosophic Oversets, Neutrosophic Undersets, Neutrosophic Offsets, and the Nonstandard Real Set. This paper defines a variety of concepts with the goal of inspiring new ideas and serving as a valuable resource for researchers in their academic pursuits.