TAKAAKI FUJITA FLORENTIN SMARANDACHE NEUTROSOPHIC SCIENCE INTERNATIONAL ASSOCIATION PUBLISHING HOUSE NSIA of and A Dynamic Survey of Soft Set Theory and Its Extensions Neutrosophic Science International Association (NSIA) Publishing House Gallup - Guayaquil United States of America – Ecuador 20 26 Takaaki Fujita, Florentin Smarandache Editor: Neutrosophic Science International Association (NSIA) Publishing House https://fs.unm.edu/NSIA/ Division of Mathematics and Sciences University of New Mexico 705 Gurley Ave., Gallup Campus NM 87301, United States of America University of Guayaquil Av. Kennedy and Av. Delta “ Dr. Salvador Allende ” University Campus Guayaquil 090514, Ecuador 1 Introduction 5 1.1 Soft Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Types of Soft Set 7 2.1 Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 HyperSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 SuperHyperSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 ( m, n ) -SuperHyperSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 TreeSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 ForestSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 IndetermSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 ContraSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9 HesiSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.10 MultiPolar Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.11 Dynamic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12 Type- n Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.13 L-Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.14 PosetSoft set (monotone soft set) . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.15 Random soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.16 Capacitary soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.17 CoverSoft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.18 FiltrationSoft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.19 T -valued soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.20 Cubic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.21 Probabilistic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.22 D-soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.23 Complex Soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.24 Real Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.25 Intersectional soft sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.26 N -soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.27 n -ary soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.28 Linguistic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.29 MetaSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Table of Content s 2.30 Double-framed Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.31 Bijective Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.32 Ranked Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.33 Refined Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.34 MultiSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.35 GraphicSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.36 CycleSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.37 ClusterSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.38 Soft Expert Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.39 Soft Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.40 Weighted Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.41 Other Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 Uncertain Soft Theory 69 3.1 Fuzzy Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Intuitionistic Fuzzy Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Neutrosophic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Plithogenic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 Uncertain Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6 Z-Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.7 Functorial Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 Applications of Soft Set 79 4.1 Soft Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Soft Topological Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Soft Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Soft Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Soft Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Soft functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7 Soft groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.8 Soft Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.9 Soft Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.10 Soft Matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.11 Soft Bitopological Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.12 Soft Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.13 Soft Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.14 Soft probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.15 Soft SemiGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.16 Soft HyperStructure and SuperHyperStructure . . . . . . . . . . . . . . . . . . . 98 4.17 Soft Graph Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.18 HyperSoft Graph Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.19 Soft Natural Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.20 Soft n -SuperHyperGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.21 Recursive Soft SuperHyperGraph . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.22 Hierarchical Soft SuperHyperGraph . . . . . . . . . . . . . . . . . . . . . . . . . 107 5 Soft Decision-Making 111 5.1 Soft decision-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2 HyperSoft TOPSIS and SuperHyperSoft TOPSIS . . . . . . . . . . . . . . . . . . 113 5.3 Soft, HyperSoft, and SuperHyperSoft AHP . . . . . . . . . . . . . . . . . . . . . 116 5.4 Soft, HyperSoft, and SuperHyperSoft VIKOR . . . . . . . . . . . . . . . . . . . . 119 6 Conclusion 123 Appendix (List of Tables) 127 4 Chapter 1 Introduction 1.1 Soft Set Theory Classical (crisp) set theory provides a precise and widely used language for formal reasoning and mathematical modeling [3]. Over the past decades, many generalized set frameworks have been introduced to represent uncertainty and vagueness, including fuzzy sets [4], intuitionistic fuzzy sets [5], hesitant fuzzy sets [6], picture fuzzy sets [7], single-valued neutrosophic sets [8, 9], quadripartitioned neutrosophic sets [10], pentapartitioned neutrosophic sets [11], double-valued neutrosophic sets [12], hesitant neutrosophic sets [13], plithogenic sets [14,15], and soft sets [2,16]. A fuzzy set assigns to each element x a single membership grade μ ( x ) ∈ [0 , 1] , thereby capturing gradual inclusion rather than a sharp yes/no decision [4,17]. Neutrosophic sets extend this view- point by associating three (generally independent) degrees T ( x ) , I ( x ) , F ( x ) ∈ [0 , 1] , interpreted as truth, indeterminacy, and falsity, respectively [8,18]. Because these models encode uncertainty more flexibly than crisp sets, they have been applied widely, for example in decision-making [19], robotics and system integration [20], artificial intelligence [21], and neural networks [22, 23]. A soft set offers a direct framework for parameterized decision modeling by associating each attribute (or parameter) with a subset of a universe, thereby handling uncertainty in a structured manner [1, 2]. Like fuzzy and neutrosophic frameworks, soft set theory has developed many extensions and variants, and its applications have been studied across a wide range of areas, including decision support and related fields. 1.2 Our Contributions In light of these developments, research on soft set theory remains important. Moreover, because a large number of papers on soft sets and their extensions continue to appear, survey-style works play an increasingly valuable role in organizing and clarifying the landscape. Motivated by this need, in this book we provide a survey-style overview of soft set theory and its major developments. 5 A Dynamic Survey of Soft Set Theory and Its Extensions Takaaki Fujita 1 ∗ and Florentin Smarandache 2 1 Independent Researcher, Tokyo, Japan. 2 Email: Takaaki.fujita060@gmail.com University of New Mexico, Gallup Campus, NM 87301, USA. Email: fsmarandache@gmail.com Abstract 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. Keywords: Soft Set, HyperSoft Set, SuperHyperSoft Set, Soft Theory Chapter 2 Types of Soft Set As types of soft sets, a wide variety of extended soft-set models have been proposed. In this chapter, we provide a survey-style introduction and brief discussion of these extensions. 2.1 Soft Set A Soft Set offers a straightforward approach to parameterized decision modeling by associating attributes (or parameters) with subsets of a universal set, effectively addressing uncertainty in a structured manner [1, 2]. Definition 2.1.1 (Soft Set) [1, 2] Let U be a universal set and A be a set of attributes. A soft set over U is a pair ( F , S ) , where S ⊆ A and F : S → P ( U ) . Here, P ( U ) denotes the power set of U . Mathematically, a soft set is represented as: ( F , S ) = { ( α, F ( α )) | α ∈ S, F ( α ) ∈ P ( U ) } Each α ∈ S is called a parameter, and F ( α ) is the set of elements in U associated with α 2.2 HyperSoft Set A HyperSoft set maps each multi-attribute value tuple to a subset of the universe, capturing combined parameter interactions [24–26]. Definition 2.2.1 (Hypersoft Set) [26] Let U be a universal set, and let A 1 , A 2 , . . . , A m be attribute domains. Define C = A 1 × A 2 × · · · × A m , the Cartesian product of these domains. A hypersoft set over U is a pair ( G, C ) , where G : C → P ( U ) . The hypersoft set is expressed as: ( G, C ) = { ( γ, G ( γ )) | γ ∈ C , G ( γ ) ∈ P ( U ) } For an m -tuple γ = ( γ 1 , γ 2 , . . . , γ m ) ∈ C , where γ i ∈ A i for i = 1 , 2 , . . . , m , G ( γ ) represents the subset of U corresponding to the combination of attribute values γ 1 , γ 2 , . . . , γ m 7 Chapter 2. Types of Soft Set Example 2.2.2 (Example of a HyperSoft Set: laptop recommendation by exact attribute tu- ples) Let U be a finite set of laptop models: U = { ` 1 , ` 2 , ` 3 , ` 4 , ` 5 } , where ` 1 = Model A, ` 2 = Model B, ` 3 = Model C, ` 4 = Model D, ` 5 = Model E. Consider m = 3 attribute domains: A 1 = { Low , Mid , High } (price tier) , A 2 = { Light , Standard } (weight class) , A 3 = { Long , Normal } (battery life) Set the hypersoft parameter domain C = A 1 × A 2 × A 3 Define a mapping G : C → P ( U ) by assigning, to each attribute tuple γ = ( γ 1 , γ 2 , γ 3 ) ∈ C , the subset G ( γ ) ⊆ U of laptops matching that exact combination. For instance, suppose the models have the following tags: model price weight battery ` 1 Low Standard Normal ` 2 Mid Light Long ` 3 Mid Standard Long ` 4 High Light Long ` 5 High Standard Normal As a concrete evaluation, take the tuple γ ∗ = ( High , Light , Long ) ∈ C Then G ( γ ∗ ) = { ` 4 } Similarly, G ( Mid , Standard , Long ) = { ` 3 } , G ( Mid , Light , Long ) = { ` 2 } Thus ( G, C ) is a HyperSoft Set over U : each parameter is a tuple of attribute values, and G ( γ ) returns the subset of objects in U that satisfy exactly that combined tuple. 2.3 SuperHyperSoft Set A SuperHyperSoft set maps tuples of subsets of attribute-value sets to universe subsets, modeling set-valued multi-attribute constraints [27–30]. 8 Chapter 2. Types of Soft Set Definition 2.3.1 (SuperHyperSoft Set) [30] Let U be a universal set, and let P ( U ) denote the power set of U . Consider n distinct attributes a 1 , a 2 , . . . , a n , where n ≥ 1 . Each attribute a i is associated with a set of attribute values A i , satisfying the property A i ∩ A j = ∅ for all i 6 = j Define P ( A i ) as the power set of A i for each i = 1 , 2 , . . . , n . Then, the Cartesian product of the power sets of attribute values is given by: C = P ( A 1 ) × P ( A 2 ) × · · · × P ( A n ) A SuperHyperSoft Set over U is a pair ( F, C ) , where: F : C → P ( U ) , and F maps each element ( α 1 , α 2 , . . . , α n ) ∈ C (with α i ∈ P ( A i ) ) to a subset F ( α 1 , α 2 , . . . , α n ) ⊆ U . Mathematically, the SuperHyperSoft Set is represented as: ( F, C ) = { ( γ, F ( γ )) | γ ∈ C , F ( γ ) ∈ P ( U ) } Here, γ = ( α 1 , α 2 , . . . , α n ) ∈ C , where α i ∈ P ( A i ) for i = 1 , 2 , . . . , n , and F ( γ ) corresponds to the subset of U defined by the combined attribute values α 1 , α 2 , . . . , α n Example 2.3.2 (Example of a SuperHyperSoft Set: meal planning with set-valued attribute choices) Let U be a set of dinner recipes: U = { r 1 , r 2 , r 3 , r 4 , r 5 , r 6 } , where r 1 = tofu salad, r 2 = chicken stir-fry, r 3 = lentil soup, r 4 = salmon bowl, r 5 = gluten-free pasta, r 6 = vegetable curry. Consider n = 3 distinct attributes: a 1 = Diet type , a 2 = Main protein , a 3 = Cooking time Let the corresponding attribute-value sets be A 1 = { Vegan , Omnivore , Pescatarian } , A 2 = { Tofu , Chicken , Fish , Legumes } , A 3 = { Quick , Medium , Long } , so A i ∩ A j = ∅ for i 6 = j . Define the super-parameter domain C = P ( A 1 ) × P ( A 2 ) × P ( A 3 ) For each γ = ( α 1 , α 2 , α 3 ) ∈ C , interpret α i ⊆ A i as a set of acceptable values for attribute a i (rather than a single value). Define F : C → P ( U ) by selecting recipes compatible with the acceptable sets. For instance, suppose the recipe tags are: recipe diet protein time r 1 Vegan Tofu Quick r 2 Omnivore Chicken Medium r 3 Vegan Legumes Long r 4 Pescatarian Fish Quick r 5 Omnivore Legumes Medium r 6 Vegan Legumes Medium 9 Chapter 2. Types of Soft Set As a concrete evaluation, take γ ∗ = ( α 1 , α 2 , α 3 ) = ( { Vegan , Pescatarian } , { Tofu , Fish } , { Quick } ) Then F ( γ ∗ ) is the subset of recipes whose diet is in α 1 , protein is in α 2 , and cooking time is in α 3 : F ( γ ∗ ) = { r 1 , r 4 } Thus ( F, C ) is a SuperHyperSoft Set: each parameter is a tuple of subsets ( α 1 , α 2 , α 3 ) specifying acceptable attribute values at each coordinate, and F ( α 1 , α 2 , α 3 ) returns the recipes satisfying the combined set-valued constraints. A comparison of soft sets, hypersoft sets, and superhypersoft sets is presented in Table 2.1. Table 2.1: Concise comparison of Soft sets, HyperSoft sets, and SuperHyperSoft sets. Aspect Soft set HyperSoft set (Hy- persoft set) SuperHyperSoft set Parameter domain A subset A ⊆ E of pa- rameters. A Cartesian product C = A 1 × · · · × A m of attribute-value do- mains. A product of powersets C = P ( A 1 ) × · · · × P ( A n ) (subset-valued attribute choices). Evaluation map (codomain) F : A → P ( U ) G : C → P ( U ) F : C → P ( U ) with C as above. Meaning of one parameter A single attribute/cri- terion e ∈ A selects a subset F ( e ) ⊆ U A full attribute-value tuple γ ∈ C selects G ( γ ) ⊆ U A tuple of value-subsets ( α 1 , . . . , α n ) ∈ C se- lects F ( α 1 , . . . , α n ) ⊆ U Typical use Parameterized selec- tion under independent criteria. Multi-attribute selec- tion under simultane- ous value assignments. Multi-attribute selec- tion under set-valued (possibly multi-choice) constraints per at- tribute. 2.4 ( m, n ) -SuperHyperSoft Set An ( m, n ) -SuperHyperSoft set parameterizes objects by m attribute groups across n hierarchical subset-levels, mapping each tuple to a universe subset. Let U be a nonempty universe, and let A 1 , A 2 , . . . , A m be m pairwise‑disjoint attribute domains. We write P ( A i ) for the power set of A i . Introduce C = m ∏ i =1 P ( A i ) = P ( A 1 ) × · · · × P ( A m ) , whose elements are tuples α = ( α 1 , . . . , α m ) with α i ⊆ A i 10 Chapter 2. Types of Soft Set Similarly, fix a single universal codomain U and consider D = n ∏ j =1 P ( U ) = P ( U ) × · · · × P ( U ) ︸ ︷︷ ︸ n factors An element of D is an n -tuple X = ( X 1 , . . . , X n ) with each X j ⊆ U Definition 2.4.1 ( ( m, n ) ‑SuperHyperSoft Set) An ( m, n ) -SuperHyperSoft Set on U (with at- tribute domains A 1 , . . . , A m ) is a function F : C −→ D Equivalently, one may write F ( α 1 , . . . , α m ) = ( F 1 ( α 1 , . . . , α m ) , . . . , F n ( α 1 , . . . , α m ) ) , where each coordinate F j : C −→ P ( U ) ( j = 1 , . . . , n ) is itself a “classical” m ‑SuperHyperSoft Set 1 Remark 2.4.2. Thus an ( m, n ) -SuperHyperSoft Set encodes n different m ‑SuperHyperSoft evaluations in parallel, one per coordinate. Example 2.4.3 (Example of an ( m, n ) -SuperHyperSoft Set: course recommendation with two- level outputs) Let U be a set of university courses: U = { c 1 , c 2 , c 3 , c 4 , c 5 , c 6 } , where c 1 = Linear Algebra, c 2 = Discrete Mathematics, c 3 = Machine Learning, c 4 = Databases, c 5 = Algorithms, c 6 = Statistics. Take m = 3 pairwise-disjoint attribute domains describing a student’s preferences: A 1 = { Math , CS , Data } (interest area) , A 2 = { Beginner , Intermediate , Advanced } (difficulty tolerance) , A 3 = { Short , Normal } (weekly workload) Define the input domain C = P ( A 1 ) × P ( A 2 ) × P ( A 3 ) Fix n = 2 output levels and define D = P ( U ) × P ( U ) Interpret the first component as recommended courses and the second as optional courses. Define F : C → D by F ( α 1 , α 2 , α 3 ) = ( F 1 ( α 1 , α 2 , α 3 ) , F 2 ( α 1 , α 2 , α 3 ) ) , 1 I.e. a mapping from C into P ( U ) 11 Chapter 2. Types of Soft Set where F 1 , F 2 : C → P ( U ) are given by a simple rule-based advisor. For a concrete parameter tuple, take ( α 1 , α 2 , α 3 ) = ( { CS , Data } , { Intermediate , Advanced } , { Normal } ) Suppose the advisor outputs F 1 ( α 1 , α 2 , α 3 ) = { c 3 , c 5 , c 4 } and F 2 ( α 1 , α 2 , α 3 ) = { c 6 , c 2 } Thus F ( α 1 , α 2 , α 3 ) = ( { c 3 , c 5 , c 4 } , { c 6 , c 2 } ) ∈ D Interpretation: the input ( α 1 , α 2 , α 3 ) specifies sets of acceptable values for each attribute group (interest area, difficulty, workload), and the output is an n -tuple of subsets of U (recommended and optional course lists). Hence F is an ( m, n ) -SuperHyperSoft Set on U with ( m, n ) = (3 , 2) For reference, the comparison between a SuperHyperSoft set and an ( m, n ) -SuperHyperSoft set is summarized in Table 2.2. Table 2.2: Concise comparison between a SuperHyperSoft set and an ( m, n ) -SuperHyperSoft set on a universe U Aspect SuperHyperSoft set (single- output) ( m, n ) -SuperHyperSoft set (multi-output) Attribute domains Pairwise-disjoint domains A 1 , . . . , A m ; each input compo- nent is a subset α i ⊆ A i Same domains A 1 , . . . , A m and the same subset-valued input components α i ⊆ A i Input (parameter) space C = ∏ m i =1 P ( A i ) Same C = ∏ m i =1 P ( A i ) Mapping (codomain) F : C → P ( U ) F : C → D , where D = ∏ n j =1 P ( U ) Output semantics For each α ∈ C , a single selected sub- set F ( α ) ⊆ U For each α ∈ C , an n -tuple F ( α ) = ( F 1 ( α ) , . . . , F n ( α )) with F j ( α ) ⊆ U (e.g., recommended/optional/rejected tiers). Equivalent viewpoint One m -attribute, subset-valued selec- tor on U An ordered family of n paral- lel selectors ( F 1 , . . . , F n ) , each an m -SuperHyperSoft-type map C → P ( U ) Reduction / relation Special case of ( m, n ) with n = 1 (identify D = P ( U ) ). Strict extension of the single-output model by allowing n coordinated out- puts for each input tuple. Typical use Set-valued multi-attribute con- straints/selection with set-valued attribute inputs. Multi-stage screening, multi-tier re- porting, or hierarchical decision out- puts under the same set-valued multi- attribute inputs. 2.5 TreeSoft Set A TreeSoft set maps subsets of a hierarchical attribute tree to universe subsets, modeling refined parameters across multiple levels [31–34]. Related concepts include PolyTree-soft sets [35] and Tree-to-Tree soft sets [36]. 12 Chapter 2. Types of Soft Set Definition 2.5.1 (TreeSoft Set) [37] Let U be a universe of discourse and let H be a nonempty subset of U . Write P ( H ) for the power set of H . Let A = { A 1 , A 2 , . . . , A n } be a set of attributes (parameters, factors, etc.), where n ≥ 1 and each A i is regarded as a first-level attribute. Each first-level attribute A i may be refined into a set of second-level sub-attributes A i = { A i, 1 , A i, 2 , . . . } Likewise, each second-level sub-attribute A i,j may be further refined into third-level sub-sub- attributes, A i,j = { A i,j, 1 , A i,j, 2 , . . . } , and so on. In general, one may consider sub-attributes at the m -th level, indexed by A i 1 ,i 2 ,...,i m , where each index i k specifies the position at level k This hierarchical attribute organization determines a rooted tree, denoted by Tree ( A ) , whose root is A (level 0 ) and whose nodes consist of all attributes and sub-attributes across levels 1 through m . The terminal nodes (nodes without descendants) are called the leaves of Tree ( A ) A TreeSoft Set on H (with attribute-tree Tree ( A ) ) is a mapping F : P ( Tree ( A ) ) −→ P ( H ) , where P ( Tree ( A )) denotes the power set of the node set of Tree ( A ) Example 2.5.2 (Example of a TreeSoft Set: medical triage rules organized by a symptom tree) Let U be a universe of patients and let H = { p 1 , p 2 , p 3 , p 4 , p 5 , p 6 } ⊆ U be a finite set of patients currently in a clinic. Consider a hierarchical attribute system with two first-level attributes: A = { A 1 , A 2 } , A 1 = “Respiratory” , A 2 = “Cardiovascular” Refine each first-level attribute into second-level sub-attributes: A 1 = { A 1 , 1 , A 1 , 2 } , A 1 , 1 = “Cough” , A 1 , 2 = “Shortness of breath” , A 2 = { A 2 , 1 , A 2 , 2 } , A 2 , 1 = “Chest pain” , A 2 , 2 = “Palpitations” Refine one second-level attribute further into third-level sub-sub-attributes: A 1 , 2 = { A 1 , 2 , 1 , A 1 , 2 , 2 } , A 1 , 2 , 1 = “Mild dyspnea” , A 1 , 2 , 2 = “Severe dyspnea” Let Tree ( A ) denote the rooted attribute tree whose nodes are Tree ( A ) = { A, A 1 , A 2 , A 1 , 1 , A 1 , 2 , A 2 , 1 , A 2 , 2 , A 1 , 2 , 1 , A 1 , 2 , 2 } Define a TreeSoft Set F : P ( Tree ( A )) −→ P ( H ) 13 Chapter 2. Types of Soft Set by mapping any chosen set of nodes X ⊆ Tree ( A ) to the subset F ( X ) ⊆ H of patients who satisfy all clinical features represented in X . Concretely, suppose the clinic records yield: F ( { A 1 , 1 } ) = { p 1 , p 2 , p 5 } (patients with cough) , F ( { A 1 , 2 , 2 } ) = { p 2 , p 6 } (patients with severe dyspnea) , F ( { A 2 , 1 } ) = { p 3 , p 6 } (patients with chest pain) For a combined node-set, define F by intersection of the corresponding patient groups; for example, F ( { A 1 , 1 , A 1 , 2 , 2 } ) = F ( { A 1 , 1 } ) ∩ F ( { A 1 , 2 , 2 } ) = { p 2 } , and F ( { A 2 , 1 , A 1 , 2 , 2 } ) = F ( { A 2 , 1 } ) ∩ F ( { A 1 , 2 , 2 } ) = { p 6 } Thus F assigns to each subset of attribute-tree nodes a subset of patients in H matching the selected hierarchical symptom description, and therefore ( F, Tree ( A )) constitutes a TreeSoft Set on H 2.6 ForestSoft Set A ForestSoft Set is formed by taking a collection of TreeSoft Sets and “gluing” (uniting) them together so as to obtain a single function whose domain is the union of all tree-nodes’ power sets and whose values in P ( H ) combine the images given by the individual TreeSoft Sets [38–41]. Definition 2.6.1 (ForestSoft Set) [40] Let U be a universe of discourse, H ⊆ U be a non-empty subset, and P ( H ) be the power set of H . Suppose we have a finite (or countable) collection of TreeSoft Sets { F t : P ( Tree ( A ( t ) )) → P ( H ) } t ∈ T , where each F t is a TreeSoft Set corresponding to a tree Tree ( A ( t ) ) of attributes A ( t ) We construct a forest by taking the (disjoint) union of all these trees: Forest ( { A ( t ) } t ∈ T ) = ⊔ t ∈ T Tree ( A ( t ) ) A ForestSoft Set , denoted by F : P ( Forest ( { A ( t ) } ) ) −→ P ( H ) , is defined as the union of all TreeSoft Set mappings F t Concretely, for any element X ∈ P ( Forest ( { A ( t ) } ) ) , we set F ( X ) = ⋃ t ∈ T X ∩ Tree ( A ( t ) ) 6 = ∅ F t ( X ∩ Tree ( A ( t ) ) ) , where we only apply F t to that portion of X belonging to the tree Tree ( A ( t ) ) 14 Chapter 2. Types of Soft Set Example 2.6.2 (Example of a ForestSoft Set: hospital triage across multiple specialty trees) Let U be a universe of patients and let H = { p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 } ⊆ U be the set of patients currently under assessment. Assume two medical specialties provide separate hierarchical attribute trees (a forest): T = { t Resp , t Card } (1) Respiratory TreeSoft Set. Let Tree ( A ( t Resp ) ) be the respiratory attribute tree with nodes Tree ( A ( t Resp ) ) = { R, R Cough , R Dyspnea , R SevDyspnea } , interpreted as R = Respiratory (root), R Cough = Cough, R Dyspnea = Dyspnea, R SevDyspnea = Severe dyspnea. Define a TreeSoft Set F t Resp : P ( Tree ( A ( t Resp ) )) → P ( H ) by patient groups: F t Resp ( { R Cough } ) = { p 1 , p 2 , p 5 } , F t Resp ( { R SevDyspnea } ) = { p 2 , p 6 , p 8 } , and (as a typical rule) for combined node-sets use intersections, e.g., F t Resp ( { R Cough , R SevDyspnea } ) = { p 1 , p 2 , p 5 } ∩ { p 2 , p 6 , p 8 } = { p 2 } (2) Cardiovascular TreeSoft Set. Let Tree ( A ( t Card ) ) be the cardiovascular attribute tree with nodes Tree ( A ( t Card ) ) = { C, C ChestPain , C Arrhythmia } , interpreted as C = Cardiovascular (root), C ChestPain = Chest pain, C Arrhythmia = Arrhyth- mia/palpitations. Define a TreeSoft Set F t Card : P ( Tree ( A ( t Card ) )) → P ( H ) by F t Card ( { C ChestPain } ) = { p 3 , p 6 } , F t Card ( { C Arrhythmia } ) = { p 4 , p 7 } , and for a combined node-set, F t Card ( { C ChestPain , C Arrhythmia } ) = { p 3 , p 6 } ∩ { p 4 , p 7 } = ∅ (3) Forest and ForestSoft Set aggregation. Form the forest by disjoint union: Forest = Tree ( A ( t Resp ) ) t Tree ( A ( t Card ) ) Define the ForestSoft Set F : P ( Forest ) → P ( H ) by F ( X ) = ⋃ t ∈ T X ∩ Tree ( A ( t ) ) 6 = ∅ F t ( X ∩ Tree ( A ( t ) ) ) For example, take the mixed selection X = { R SevDyspnea , C ChestPain } ⊆ Forest Then F ( X ) = F t Resp ( { R SevDyspnea } ) ∪ F t Card ( { C ChestPain } ) = { p 2 , p 6 , p 8 }∪{ p 3 , p 6 } = { p 2 , p 3 , p 6 , p 8 } Interpretation: the ForestSoft Set aggregates the (possibly different) specialty-specific TreeSoft Set outputs, enabling a unified view across multiple hierarchical symptom trees. 15 Chapter 2. Types of Soft Set 2.7 IndetermSoft Set Single-valued IndetermSoft Set maps each attribute value to one subset capturing undirected, non-unique indeterminacy over H; domain/codomain may be indeterminate [42–46]. Definition 2.7.1 ((Single-valued) IndetermSoft set) [24, 37, 47, 48] Let U be a universe of discourse, H ⊆ U a non-empty subset, and P ( H ) the powerset of H Let A be the set of attribute values for an attribute a . A function F : A → P ( H ) is called an IndetermSoft Set if at least one of the following conditions holds: 1. A has some indeterminacy. 2. P ( H ) has some indeterminacy. 3. There exists at least one v ∈ A such that F ( v ) is indeterminate (unclear, uncertain, or not unique). 4. Any two or all three of the above conditions. An IndetermSoft Set is represented mathematically as: F : A → H ( ∩ , ∪ , ⊕ , ¬ ) , where H ( ∩ , ∪ , ⊕ , ¬ ) represents a structure closed under the IndetermSoft operators. Example 2.7.2 (Example of a (single-valued) IndetermSoft set: recruiting with missing/uncer- tain evidence) Let U be the set of shortlisted applicants for a data-engineering position: U = { u 1 , u 2 , u 3 , u 4 , u 5 } , H := U. Consider one attribute a = “technical screening outcome” with a value-set A = { Pass , Borderline , Fail } Define F : A → P ( H ) as follows. Suppose the company has completed the screening, but two applicants ( u 2 , u 5 ) have indeterminate results due to missing logs and a disputed proctoring report. Thus, the subsets corresponding to clear outcomes are: F ( Pass ) = { u 1 , u 3 } , F ( Fail ) = { u 4 } , while the “Borderline” group is not uniquely determined : depending on which audit is accepted, either u 2 is borderline and u 5 is cleared, or vice versa. Hence we treat F ( Borderline ) = indeterminate (not unique) One convenient single-valued representation is to regard F ( Borderline ) as taking values in a family of possible subsets (an indeterminate value), e.g., F ( Borderline ) ∈ { { u 2 } , { u 5 } } Interpretation: the attribute-value set A is crisp, the universe H is crisp, but at least one value F ( v ) (here v = Borderline) is indeterminate/unclear/not unique . Therefore F satisfies Condi- tion (3) in Definition (Single-valued) IndetermSoft set , and so F constitutes an IndetermSoft set on H 16 Chapter 2. Types of Soft Set Related concepts include the following notions. • IndetermHyperSoft Set [43, 49, 50]: Hypersoft set whose parameter tuples or images may be indeterminate, nonunique, or partially specified. • IndetermSuperHyperSoft Set [51]: Superhypersoft set allowing indeterminacy in higher- order parameter subsets and corresponding approximations. • Bipolar IndetermSoft Set [52]: Indetermsoft set with positive and negative evaluations, permitting indeterminacy within both perspectives. • Weighted Indetermsoft set [44]: Weighted IndetermSoft set assigns each attribute a weight and indeterminate approximation, enabling prioritized decision-making under uncertainty with incomplete data often. 2.8 ContraSoft Set A ContraSoft Set is a parameterized soft set where each parameter’s values are associated with a contradiction degree, and thresholding is used to aggregate only those values that are not too contradictory with respect to a chosen reference [53]. This allows soft-set modeling to filter or weight information based on contradiction, rather than uncertainty. Definition 2.8.1 (Contradiction on attribute values) [53] Let V be a nonempty finite set of attribute values. A contradiction function on V is a map c : V × V −→ [0 , 1] such that c ( v, v ) = 0 ( reflexivity ) , c ( v, w ) = c ( w, v ) ( symmetry ) The quantity c ( v, w ) measures the degree of contradiction between v and w (larger means more contradictory). Definition 2.8.2 (ContraSoft structure) Let U be a nonempty universe and E a nonempty set of parameters. For each e ∈ E fix: • a nonempty finite value set V e ; • a contradiction function c e : V e × V e → [0 , 1] (Definition 2.8.1); • a designated reference value v ? e ∈ V e Write V := ⊔ e ∈ E ( { e } × V e ) for the disjoint union of all parameter–value pairs. 17 Chapter 2. Types of Soft Set Definition 2.8.3 (ContraSoft Set) Let U be a finite universe of objects and E a finite set of parameters. A ContraSoft Set is a quadruple CS := ( U, E, F, c ) , where • F : E → P ( U ) is the (crisp) soft mapping; F ( e ) ⊆ U is the set of objects accepted (or classified as positive) under parameter e ; • c : E × E → [0 , 1] is a contradiction degree on parameters, symmetric and reflexive on the diagonal: c ( e, e ) = 0 , c ( e, f ) = c ( f, e ) ( ∀ e, f ∈ E ) For x ∈ U and e ∈ E , the atomic lemma “ x is accepted by e ” is represented by A ( x, e ) : x ∈ F ( e ) , with truth value T if x ∈ F ( e ) and F otherwise. Remark 2.8.4 (Relation to classical soft sets and to “indeterminacy”) If V e = { v ? e } for all e , then F ( τ ) ( e ) = F ( e, v ? e ) and we recover the classical soft set ( F ◦ , E ) with F ◦ ( e ) = F ( e, v ? e ) Thus, contradiction plays the role of the third component often used as “indeterminacy” (e.g. in neutrosophic settings), but here it acts as a distance-to-reference that controls which value-slices are admitted into F ( τ ) ( e ) Example 2.8.5 (Real-life example of a ContraSoft Set: hiring filters with contradictory criteria) Let U be a finite set of job applicants: U = { u 1 , u 2 , u 3 , u 4 , u 5 , u 6 } Let E be a finite set of screening parameters: E = { e Exp , e Salary , e Remote , e Onsite } , where e Exp = “has strong experience”, e Salary = “fits low salary budget”, e Remote = “prefers fully remote”, and e Onsite = “prefers on-site”. Define the soft mapping F : E → P ( U ) by the applicants accepted under each criterion: F ( e Exp ) = { u 1 , u 3 , u 5 } , F ( e Salary ) = { u 2 , u 4 , u 6 } , F ( e Remote ) = { u 1 , u 2 , u 6 } , F ( e Onsite ) = { u 3 , u 4 , u 5 } Now define a contradiction degree c : E × E → [0 , 1] capturing how incompatible two parameters are. For example, “remote” and “on-site” are highly contradictory, while “experience” and “salary budget” are moderately contradictory: c ( e Remote , e Onsite ) = c ( e Onsite , e Remote ) = 0 95 , c ( e Exp , e Salary ) = c ( e Salary , e Exp ) = 0 60 , and set c ( e, e ) = 0 for all e ∈ E ; for all other unordered pairs not listed above, take c = 0 20 Then CS = ( U, E, F, c ) is a ContraSoft Set. Interpretation: when aggregating decisions across parameters, one may downweight or discard simultaneously using highly contradictory criteria (e.g., combining e Remote with e Onsite ), while allowing combinations with low contradiction. A comparison between Soft Sets and ContraSoft Sets is presented in Table 2.3. 18 Chapter 2. Types of Soft Set Table 2.3: Soft Set vs. ContraSoft Set (concise comparison) Aspect Soft Set ContraSoft Set Core idea Parameterized family of subsets of a universe. Soft Set augmented with contra- diction degrees to control accep- tance/weighting. Universe/Parame- ters Universe U , parameter set E Same U, E plus contradiction map(s). Mapping F : E → P ( U ) F : E → P ( U ) together with contradiction c on parameters and/or values. Extra structure None. c : E × E → [0 , 1] (and optionally c e : V e × V e → [0 , 1] ), reference(s), tolerance τ Selection / aggrega- tion Set-theoretic filtering (union, in- tersection) across parameters. Contradiction-aware filtering F ( τ ) and/or weighted aggregation (plithogenic-style). Typical use Parameter-driven modeling of un- certainty and preferences. Conflict-aware modeling when pa- rameters/values may be mutually opposing. Reduction — Recovers Soft Set when c ≡ 0 (and no contradiction-based fil- tering is applied). 2.9 HesiSoft Set A HesiSoft Set is a soft set F : E → P ( U ) together with a symmetric hesitancy map h : E × E → P fin ([0 , 1]) . Related concepts include hesitant fuzzy sets [6,54] and hesitant neutrosophic sets [13, 55]. Definition 2.9.1 (HesiSoft Set) Let U be a finite universe of objects and E a finite set of parameters. A HesiSoft Set is a quadruple HSS := ( U, E, F, h ) , where • F : E → P ( U ) is the (crisp) soft mapping; F ( e ) ⊆ U is the set of objects accepted (or classified as positive) under parameter e ; • h : E × E → P fin ([0 , 1]) is a hesitancy map on parameters, symmetric and normalized on the diagonal: h ( e, e ) = { 0 } , h ( e, f ) = h ( f, e ) ( ∀ e, f ∈ E ) , where P fin ([0 , 1]) denotes the family of all finite subsets of [0 , 1] For x ∈ U and e ∈ E , the atomic statement “ x is accepted by e ” is represented by A ( x, e ) : x ∈ F ( e ) , with truth value T if x ∈ F ( e ) and F otherwise. 19