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A Dynamic Survey of Soft Set Theory and Its Extensions
Takaaki Fujita, Florentin Smarandache
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Summary
This paper presents a comprehensive survey of soft set theory and its numerous extensions, including HyperSoft, SuperHyperSoft, TreeSoft, and Dynamic Soft sets. It details the mathematical definitions, hierarchical structures, and parameter mappings of these variants, while also exploring their applications in topology, algebra, decision-making (TOPSIS, AHP, VIKOR), and artificial intelligence (neural networks, graph theory).
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Relation Signals (9)
Takaaki Fujita → authored → A Dynamic Survey of Soft Set Theory and Its Extensions
confidence 99% · Takaaki Fujita, Florentin Smarandache Editor
Florentin Smarandache → authored → A Dynamic Survey of Soft Set Theory and Its Extensions
confidence 99% · Takaaki Fujita, Florentin Smarandache Editor
Soft Set → isbasefor → HyperSoft Set
confidence 95% · Over the past decades, the theory has expanded into numerous variants-including hypersoft sets
HyperSoft Set → isextensionof → Soft Set
confidence 95% · A HyperSoft set maps each multi-attribute value tuple to a subset of the universe
SuperHyperSoft Set → isextensionof → HyperSoft Set
confidence 92% · A SuperHyperSoft set maps tuples of subsets of attribute-value sets to universe subsets
Soft Set → usedin → TOPSIS
confidence 88% · 5.2 HyperSoft TOPSIS and SuperHyperSoft TOPSIS
Soft Set → usedin → AHP
confidence 88% · 5.3 Soft, HyperSoft, and SuperHyperSoft AHP
Soft Set → usedin → VIKOR
confidence 88% · 5.4 Soft, HyperSoft, and SuperHyperSoft VIKOR
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Abstract
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.
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- Source: https://arxiv.org/abs/2602.21268v1
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TAKAAKI FUJITA FLORENTIN SMARANDACHE NEUTROSOPHIC SCIENCE INTERNATIONAL ASSOCIATION PUBLISHING HOUSE NSIA of and A Dynamic Survey of Soft Set Theory and Its Extensions Neutrosophic Scien ce International Association (NSIA) Publis hing House Gallup - Guayaquil United States of America – Ecuador 2026 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 Univ ersity of Guayaquil A v. Kennedy and Av. Delta “Dr. Salvador Allende” University Campus Guayaquil 090514, Ecuador 1 Introduction5 1.1 Soft Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Types of Soft Set7 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-nSoft 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 2.19T-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 2.26N-soft Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.27n-ary soft set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.28 Linguistic Soft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.29 MetaSoft Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Table of Contents 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 Theory69 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 Set79 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 Softn-SuperHyperGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.21 Recursive Soft SuperHyperGraph . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.22 Hierarchical Soft SuperHyperGraph . . . . . . . . . . . . . . . . . . . . . . . . . 107 5 Soft Decision-Making111 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 Conclusion123 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 elementxa single membership gradeμ(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) degreesT(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] LetUbe a universal set andAbe a set of attributes. A soft set overUis a pair(F, S), whereS⊆AandF:S→ P(U). Here,P(U)denotes the power set ofU. Mathematically, a soft set is represented as: (F, S) =(α,F(α))|α∈S,F(α)∈P(U). Eachα∈Sis called a parameter, andF(α)is the set of elements inUassociated withα. 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] LetUbe a universal set, and letA 1 ,A 2 , . . . ,A m be attribute domains. DefineC=A 1 ×A 2 ×·×A m , the Cartesian product of these domains. A hypersoft set overUis a pair(G,C), whereG:C →P(U). The hypersoft set is expressed as: (G,C) =(γ, G(γ))|γ∈C, G(γ)∈P(U). For anm-tupleγ= (γ 1 , γ 2 , . . . , γ m )∈C, whereγ i ∈A i fori= 1,2, . . . , m,G(γ)represents the subset ofUcorresponding to the combination of attribute valuesγ 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).LetUbe 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. Considerm= 3attribute 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 mappingG:C →P(U)by assigning, to each attribute tupleγ= (γ 1 , γ 2 , γ 3 )∈C, the subsetG(γ)⊆Uof laptops matching that exact combination. For instance, suppose the models have the following tags: model price weight battery ` 1 Low Standard Normal ` 2 Mid LightLong ` 3 Mid Standard Long ` 4 High LightLong ` 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 overU: each parameter is atupleof attribute values, andG(γ) returns the subset of objects inUthat 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] LetUbe a universal set, and letP(U)denote the power set ofU. Considerndistinct attributesa 1 , a 2 , . . . , a n , wheren≥1. Each attributea i is associated with a set of attribute valuesA i , satisfying the propertyA i ∩A j =∅for alli6=j. DefineP(A i )as the power set ofA i for eachi= 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 overUis a pair(F,C), where: F:C →P(U), andFmaps each element(α 1 , α 2 , . . . , α n )∈C(withα i ∈P(A i )) to a subsetF(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, whereα i ∈P(A i )fori= 1,2, . . . , n, andF(γ)corresponds to the subset ofUdefined by the combined attribute valuesα 1 , α 2 , . . . , α n . Example 2.3.2(Example of a SuperHyperSoft Set: meal planning with set-valued attribute choices).LetUbe a set of dinner recipes: U=r 1 , r 2 , r 3 , r 4 , r 5 , r 6 , wherer 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. Considern= 3distinct 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, soA i ∩A j =∅fori6 =j. Define the super-parameter domain C=P(A 1 )×P(A 2 )×P(A 3 ). For eachγ= (α 1 , α 2 , α 3 )∈ C, interpretα i ⊆A i as aset of acceptable valuesfor attributea i (rather than a single value). DefineF:C → P(U)by selecting recipes compatible with the acceptable sets. For instance, suppose the recipe tags are: recipedietproteintime r 1 VeganTofuQuick r 2 Omnivore Chicken Medium r 3 VeganLegumes Long r 4 Pescatarian FishQuick r 5 Omnivore Legumes Medium r 6 VeganLegumes 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). ThenF(γ ∗ )is the subset of recipes whose diet is inα 1 , protein is inα 2 , and cooking time is in α 3 : F(γ ∗ ) =r 1 , r 4 . Thus(F,C)is a SuperHyperSoft Set: each parameter is atuple of subsets(α 1 , α 2 , α 3 )specifying acceptable attribute values at each coordinate, andF(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. AspectSoft setHyperSoft set (Hy- persoft set) SuperHyperSoft set Parameter domainA subsetA⊆Eof pa- rameters. A Cartesian product C=A 1 × · × A m of attribute-value do- mains. A product ofpowersets 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)withC as above. Meaning of one parameter A single attribute/cri- terione∈Aselects a subsetF(e)⊆U. A full attribute-value tupleγ∈ Cselects G(γ)⊆U. A tuple ofvalue-subsets (α 1 , . . . , α n )∈ Cse- lectsF(α 1 , . . . , α n )⊆ U. Typical useParameterized selec- tion under independent criteria. Multi-attribute selec- tion under simultane- ous value assignments. Multi-attribute selec- tion underset-valued (possibly multi-choice) constraints per at- tribute. 2.4(m,n)-SuperHyperSoft Set An(m, n)-SuperHyperSoft set parameterizes objects bymattribute groups acrossnhierarchical subset-levels, mapping each tuple to a universe subset. LetUbe a nonempty universe, and let A 1 , A 2 , . . . , A m bempairwise‑disjoint attribute domains. We writeP(A i )for the power set ofA i . Introduce C= m ∏ i=1 P(A i ) =P(A 1 )×·×P(A m ), whose elements are tuplesα= (α 1 , . . . , α m )withα i ⊆A i . 10 Chapter 2. Types of Soft Set Similarly, fix a single universal codomainUand consider D= n ∏ j=1 P(U) =P(U)×·×P(U) ︸︷︸ nfactors . An element ofDis ann-tupleX= (X 1 , . . . , X n )with eachX j ⊆U. Definition 2.4.1((m, n)‑SuperHyperSoft Set).An(m, n)-SuperHyperSoft SetonU(with at- tribute domainsA 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 encodesndifferentm‑SuperHyperSoft evaluations in parallel, one per coordinate. Example 2.4.3(Example of an(m, n)-SuperHyperSoft Set: course recommendation with two- level outputs).LetUbe a set of university courses: U=c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , wherec 1 = Linear Algebra,c 2 = Discrete Mathematics,c 3 = Machine Learning,c 4 = Databases, c 5 = Algorithms,c 6 = Statistics. Takem= 3pairwise-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 ). Fixn= 2output levels and define D=P(U)×P(U). Interpret the first component asrecommendedcourses and the second asoptionalcourses. DefineF:C →Dby 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 fromCintoP(U). 11 Chapter 2. Types of Soft Set whereF 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 andF 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 )specifiessetsof acceptable values for each attribute group (interest area, difficulty, workload), and the output is ann-tuple of subsets ofU(recommended and optional course lists). HenceFis an(m, n)-SuperHyperSoft Set onUwith(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 universeU. AspectSuperHyperSoft set (single- output) (m, n)-SuperHyperSoftset (multi-output) Attribute domainsPairwise-disjointdomains A 1 , . . . , A m ; each input compo- nent is a subsetα i ⊆A i . Same domainsA 1 , . . . , A m and the same subset-valued input components α i ⊆A i . Input (parameter) spaceC= ∏ m i=1 P(A i ).SameC= ∏ m i=1 P(A i ). Mapping (codomain)F:C →P(U).F:C →D, whereD= ∏ n j=1 P(U). Output semanticsFor eachα∈C, a single selected sub- setF(α)⊆U. For eachα∈ C, ann-tupleF(α) = (F 1 (α), . . . , F n (α))withF j (α)⊆U (e.g., recommended/optional/rejected tiers). Equivalent viewpointOnem-attribute, subset-valued selec- tor onU. An ordered family ofnparal- lel selectors(F 1 , . . . , F n ), each an m-SuperHyperSoft-type mapC → P(U). Reduction / relationSpecial case of(m, n)withn= 1 (identifyD=P(U)). Strict extension of the single-output model by allowingncoordinated out- puts for each input tuple. Typical useSet-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 includePolyTree-soft sets[35] and Tree-to-Tree soft sets[36]. 12 Chapter 2. Types of Soft Set Definition 2.5.1(TreeSoft Set).[37] LetUbe a universe of discourse and letHbe a nonempty subset ofU. WriteP(H)for the power set ofH. LetA=A 1 , A 2 , . . . , A n be a set of attributes (parameters, factors, etc.), wheren≥1and eachA i is regarded as afirst-levelattribute. Each first-level attributeA i may be refined into a set ofsecond-levelsub-attributes A i =A i,1 , A i,2 , . . .. Likewise, each second-level sub-attributeA i,j may be further refined intothird-levelsub-sub- attributes, A i,j =A i,j,1 , A i,j,2 , . . ., and so on. In general, one may consider sub-attributes at them-th level, indexed byA i 1 ,i 2 ,...,i m , where each indexi k specifies the position at levelk. This hierarchical attribute organization determines a rooted tree, denoted by Tree(A), whose root isA(level0) and whose nodes consist of all attributes and sub-attributes across levels1 throughm. The terminal nodes (nodes without descendants) are called theleavesof Tree(A). ATreeSoft SetonH(with attribute-tree Tree(A)) is a mapping F:P ( Tree(A) ) −→P(H), whereP(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). LetUbe 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 nodesX⊆Tree(A)to the subsetF(X)⊆Hof patients who satisfyallclinical features represented inX. 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, defineFby 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 . ThusFassigns to each subset of attribute-tree nodes a subset of patients inHmatching the selected hierarchical symptom description, and therefore(F,Tree(A))constitutes a TreeSoft Set onH. 2.6 ForestSoft Set AForestSoft Setis 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 inP(H)combine the images given by the individual TreeSoft Sets [38–41]. Definition 2.6.1(ForestSoft Set).[40] LetUbe a universe of discourse,H⊆Ube a non-empty subset, andP(H)be the power set ofH. Suppose we have a finite (or countable) collection of TreeSoft Sets F t :P(Tree(A (t) ))→P(H) t∈T , where eachF t is a TreeSoft Set corresponding to a tree Tree(A (t) )of attributesA (t) . We construct aforestby taking the (disjoint) union of all these trees: Forest ( A (t) t∈T ) = ⊔ t∈T Tree ( A (t) ) . AForestSoft Set, denoted by F:P ( Forest(A (t) ) ) −→P(H), is defined as theunionof all TreeSoft Set mappingsF t . Concretely, for any elementX∈ P ( Forest(A (t) ) ) , we set F(X) = ⋃ t∈T X∩Tree(A (t) )6 =∅ F t ( X∩Tree(A (t) ) ) , where we only applyF t to that portion ofXbelonging 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). LetUbe 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 provideseparatehierarchical 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 asR=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 asC=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 ) )tTree(A (t Card ) ). Define the ForestSoft SetF: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] LetUbe a universe of discourse,H⊆Ua non-empty subset, andP(H)the powerset ofH. LetAbe the set of attribute values for an attributea. A functionF:A→P(H)is called anIndetermSoft Setif at least one of the following conditions holds: 1.Ahas some indeterminacy. 2.P(H)has some indeterminacy. 3. There exists at least onev∈Asuch thatF(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(∩,∪,⊕,¬), whereH(∩,∪,⊕,¬)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).LetUbe 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 attributea=“technical screening outcome” with a value-set A=Pass,Borderline,Fail. DefineF:A→ P(H)as follows. Suppose the company has completed the screening, but two applicants (u 2 , u 5 ) haveindeterminateresults 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 isnot uniquely determined: depending on which audit is accepted, eitheru 2 is borderline andu 5 is cleared, or vice versa. Hence we treat F(Borderline) =indeterminate (not unique). One convenient single-valued representation is to regardF(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 setAis crisp, the universeHis crisp, but at least one value F(v)(herev=Borderline) isindeterminate/unclear/not unique. ThereforeFsatisfies Condi- tion (3) in Definition(Single-valued) IndetermSoft set, and soFconstitutes an IndetermSoft set onH. 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] LetVbe a nonempty finite set of attribute values. Acontradiction functiononVis a map c:V×V−→[0,1] such that c(v, v) = 0 (reflexivity),c(v, w) =c(w, v) (symmetry). The quantityc(v, w)measures the degree ofcontradictionbetweenvandw(larger means more contradictory). Definition 2.8.2(ContraSoft structure).LetUbe a nonempty universe andEa nonempty set of parameters. For eache∈Efix: • a nonempty finite value setV e ; • a contradiction functionc e :V e ×V e →[0,1](Definition 2.8.1); • a designatedreference valuev ? e ∈V e . WriteV:= ⊔ 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).LetUbe a finite universe of objects andEa finite set of parameters. AContraSoft Setis a quadruple CS:= (U, E, F, c), where •F:E→ P(U)is the (crisp) soft mapping;F(e)⊆Uis the set of objectsaccepted(or classified as positive) under parametere; •c:E×E→[0,1]is acontradiction degreeon parameters, symmetric and reflexive on the diagonal: c(e, e) = 0,c(e, f) =c(f, e) (∀e, f∈E). Forx∈Uande∈E, the atomic lemma “xis accepted bye” is represented by A(x, e) :x∈F(e), with truth valueTifx∈F(e)andFotherwise. Remark 2.8.4(Relation to classical soft sets and to “indeterminacy”).IfV e =v ? e for alle, thenF (τ) (e) =F(e, v ? e )and we recover the classical soft set(F ◦ , E)withF ◦ (e) =F(e, v ? e ). Thus,contradictionplays the role of the third component often used as “indeterminacy” (e.g. in neutrosophic settings), but here it acts as adistance-to-referencethat controls which value-slices are admitted intoF (τ) (e). Example 2.8.5(Real-life example of a ContraSoft Set: hiring filters with contradictory criteria). LetUbe a finite set of job applicants: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . LetEbe a finite set of screening parameters: E=e Exp , e Salary , e Remote , e Onsite , wheree Exp =“has strong experience”,e Salary =“fits low salary budget”,e Remote =“prefers fully remote”, ande Onsite =“prefers on-site”. Define the soft mappingF: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 degreec: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 setc(e, e) = 0for alle∈E; for all other unordered pairs not listed above, takec= 0.20. ThenCS= (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., combininge Remote withe 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) AspectSoft SetContraSoft Set Core ideaParameterized family of subsets of a universe. Soft Set augmented with contra- diction degrees to control accep- tance/weighting. Universe/Parame- ters UniverseU, parameter setE.SameU, Eplus contradiction map(s). MappingF:E→P(U).F:E→ P(U)together with contradictioncon parameters and/or values. Extra structureNone.c:E×E→[0,1](and optionally c e :V e ×V e →[0,1]), reference(s), toleranceτ. Selection / aggrega- tion Set-theoretic filtering (union, in- tersection) across parameters. Contradiction-awarefiltering F (τ) and/or weighted aggregation (plithogenic-style). Typical useParameter-driven modeling of un- certainty and preferences. Conflict-aware modeling when pa- rameters/values may be mutually opposing. Reduction—Recovers Soft Set whenc≡0 (and no contradiction-based fil- tering is applied). 2.9 HesiSoft Set AHesiSoft Setis a soft setF:E→ P(U)together with a symmetric hesitancy maph: E×E→P fin ([0,1]). Related concepts includehesitant fuzzy sets[6,54] andhesitant neutrosophic sets[13,55]. Definition 2.9.1(HesiSoft Set).LetUbe a finite universe of objects andEa finite set of parameters. AHesiSoft Setis a quadruple HSS:= (U, E, F, h), where •F:E→ P(U)is the (crisp) soft mapping;F(e)⊆Uis the set of objectsaccepted(or classified as positive) under parametere; •h:E×E→P fin ([0,1])is ahesitancy mapon parameters, symmetric and normalized on the diagonal: h(e, e) =0,h(e, f) =h(f, e) (∀e, f∈E), whereP fin ([0,1])denotes the family of all finite subsets of[0,1]. Forx∈Uande∈E, the atomic statement “xis accepted bye” is represented by A(x, e) :x∈F(e), with truth valueTifx∈F(e)andFotherwise. 19 Chapter 2. Types of Soft Set Example 2.9.2(Hiring shortlisting with parameter-wise hesitancy).LetUbe the set of appli- cants U=u 1 , u 2 , u 3 , u 4 , u 5 , and letEbe the set of evaluation parameters E=Tech,Comm,Lead,Culture, standing for technical skills, communication, leadership, and culture fit, respectively. (1) Crisp acceptance map.DefineF:E→P(U)by F(Tech) =u 1 , u 2 , u 4 , F(Comm) =u 2 , u 3 , u 5 , F(Lead) =u 1 , u 3 , F(Culture) =u 2 , u 4 , u 5 . Thus, for example,A(u 4 ,Tech)is true (sinceu 4 ∈F(Tech)), whileA(u 4 ,Comm)is false (since u 4 /∈F(Comm)). (2) Hesitancy map on parameters.Defineh:E×E→P fin ([0,1])by settingh(e, e) =0 for alle∈E, and for distinct parameters specify (symmetrically): h(Tech,Comm) =0.2,0.4,h(Tech,Lead) =0.1,0.3, h(Tech,Culture) =0.4,0.6, h(Comm,Lead) =0.3,0.5, h(Comm,Culture) =0.1,0.2, h(Lead,Culture) =0.5,0.7, andh(e, f) =h(f, e)for all pairs. Here each finite seth(e, f)records acommittee hesitancy profileabout how strongly the two criteriaeandfshould co-influence a final hiring decision (e.g., disagreements or multiple plausible weights coming from different interviewers). Then HSS= (U, E, F, h) is a HesiSoft Set modeling a real hiring shortlist:Fcaptures crisp accept/reject outcomes per criterion, whilehcaptures finite-valued hesitancy between criteria induced by mixed expert opinions. 2.10 MultiPolar Soft Set A multipolar soft set assigns to each parameter multiple “polar” subsets, thereby capturing several perspectives or evaluations over the same universe. Related notions include multipolar fuzzy sets [56] and multipolar neutrosophic sets [57–59]. Definition 2.10.1(Multipolar Soft Set).LetUbe a nonempty universe and letEbe a nonempty set of parameters. Fix an integerm≥2(the number of poles). Anm-polar (multipolar) soft set overUwith respect toEis an ordered pair(F, E), where F:E−→ ( P(U) ) m 20 Chapter 2. Types of Soft Set is a mapping. For each parametere∈E, we write F(e) = ( F 1 (e), F 2 (e), . . . , F m (e) ) , where each component satisfiesF i (e)⊆Ufori= 1,2, . . . , m. The subsetF i (e)is called the i-th polar approximationofUunder the parametere, and it represents the evaluation ofefrom thei-th perspective (pole). Equivalently, anm-polar soft set can be viewed as an ordered family of ordinary soft sets(F 1 , E), . . . ,(F m , E)on the same universe and the same parameter set. Remark 2.10.2.Ifm= 1, thenF:E→P(U)and(F, E)reduces to an ordinary (crisp) soft set. Form= 2, the model yields a two-pole soft representation (often studied as a bipolar-type framework, possibly with additional constraints depending on the chosen bipolar definition). Example 2.10.3(Real-life example of a multipolar soft set: multi-stakeholder project risk screening).LetUbe a set of candidate IT projects in a company: U=p 1 , p 2 , p 3 , p 4 , p 5 . LetEbe a set of risk-related parameters and take E=e Sec , e Cost , e Sched , wheree Sec =“security risk”,e Cost =“budget risk”,e Sched =“schedule risk”. Suppose three different stakeholder groups evaluate each risk parameter: m= 3,(1) Security team, (2) Finance team, (3) PMO. Define a mappingF:E→ ( P(U) ) 3 by letting, for each parametere∈E, F(e) = ( F 1 (e), F 2 (e), F 3 (e) ) , whereF i (e)⊆Uis the set of projects judged by stakeholderito havehighrisk undere. For instance, assume the following assessments: F(e Sec ) = ( p 2 , p 4 ,p 4 ,p 2 , p 3 , p 4 ) , F(e Cost ) = ( p 3 ,p 1 , p 3 , p 5 ,p 1 , p 5 ) , F(e Sched ) = ( p 2 , p 5 ,p 5 ,p 1 , p 2 , p 5 ) . Then(F, E)is a3-polar (multipolar) soft set overU: for each risk parametere, the three components represent the “high-risk” project subsets identified from three distinct perspectives. Interpretation: this structure supports decisions such asconsensus high riskfore(intersection ⋂ 3 i=1 F i (e)), orany-stakeholder high risk(union ⋃ 3 i=1 F i (e)), depending on the organization’s risk policy. Related notions include the following concepts. •Bipolar Soft Sets[60–62]: model parameters by paired positive/negative approximations, enabling simultaneous support and opposition assessments for each object. •Bipolar HyperSoft Sets[63–65]: extend bipolar soft sets to multi-attribute tuple parameters, assigning positive and negative approximations to each tuple. •HyperPolar Soft Sets[66]: A hyperpolar soft set assigns each parameternpolarity-based subsets of the universe, capturing qualitative evaluations from multiple agents simultane- ously. 21 Chapter 2. Types of Soft Set 2.11 Dynamic Soft Set A dynamic soft set is a time-indexed family of soft sets, modeling parameterized approximations that evolve across time or contexts [67–69]. Definition 2.11.1(Dynamic soft set).[67–69] LetUbe a nonempty universe of discourse, let Ebe a (nonempty) set of parameters, and letTbe a nonempty index set (e.g., time points, system states, or contexts). Adynamic soft setover(U, E)indexed byTis a family S=(t, F t , A t )|t∈T, such that for eacht∈T: A t ⊆EandF t :A t −→P(U). Equivalently, one may representSas the set of triples S=(t, e, F t (e))|t∈T, e∈A t , whereF t (e)⊆Uis the (crisp) approximation ofUwith respect to parametereat indext. For eacht∈T, the pair(F t , A t )is thetime-/context-slice soft setofSatt. Remark 2.11.2(Reduction to a classical soft set).If there exists a fixedA⊆Eand a fixed mappingF:A→P(U)such thatA t =AandF t =Ffor allt∈T, then the dynamic soft set Sreduces to the classical (static) soft set(F, A). Example 2.11.3(Real-life example of a dynamic soft set: daily product availability in a grocery store).LetUbe a set of products sold by a grocery store: U=p 1 , p 2 , p 3 , p 4 , p 5 , p 6 . LetEbe a set of availability-related parameters: E=e In , e Sale , e Out , wheree In =“in stock”,e Sale =“on sale”, ande Out =“out of stock”. Let the time index set be three consecutive days, T=t 1 , t 2 , t 3 . For each dayt∈T, define a time-slice soft set(F t , A t )describing the store status. Take A t =e In , e Sale , e Out ⊆Efor allt∈T. Dayt 1 : F t 1 (e In ) =p 1 , p 2 , p 3 , p 5 , F t 1 (e Sale ) =p 2 , p 5 , F t 1 (e Out ) =p 4 , p 6 . 22 Chapter 2. Types of Soft Set Dayt 2 : F t 2 (e In ) =p 1 , p 3 , p 4 , p 5 , F t 2 (e Sale ) =p 1 , p 4 , F t 2 (e Out ) =p 2 , p 6 . Dayt 3 : F t 3 (e In ) =p 1 , p 2 , p 4 , p 6 , F t 3 (e Sale ) =p 2 , p 6 , F t 3 (e Out ) =p 3 , p 5 . Define S=(t, F t , A t )|t∈T. ThenSis a dynamic soft set over(U, E)indexed byT: each time pointthas its own soft mappingF t :A t → P(U)describing which products are in stock, on sale, or out of stock on that day. Interpretation: the same parameter (e.g., “in stock”) may select different subsets of products as inventory changes over time, and the dynamic soft set records these time-dependent approxima- tions. 2.12 Type-nSoft Set A Type-nsoft set iteratively parameterizes soft sets, assigning each parameter a Type-(n−1)soft set over the universe. Related concepts include Type-2 fuzzy sets [70,71], Type-2 neutrosophic sets [72–74], and Type-2 soft sets [75]. Definition 2.12.1(Type-nsoft set).LetUbe a nonempty universe and letEbe a nonempty set of parameters. Define, recursively, the classesΣ (n) (U, E)of Type-nsoft sets over(U, E)as follows. (1)Type-1 (classical) soft sets. Σ (1) (U, E) := (F, A) ∣ ∣ A⊆E, F:A→P(U) . Elements ofΣ (1) (U, E)are (crisp) soft sets in the sense of Molodtsov. (2)Inductive step.For each integern≥2, define Σ (n) (U, E) := (F, A) ∣ ∣ A⊆E, F:A→Σ (n−1) (U, E) . An element(F, A)∈Σ (n) (U, E)is called aType-nsoft set(briefly,T n S) over(U, E). The set Ais theprimary parameter set. For eacha∈Awe may write F(a) = (F a , A a )∈Σ (n−1) (U, E), so thatA a ⊆Eis a (possiblya-dependent)underlying parameter set at level2. Iterating this decomposition, for any chain a 1 ∈A, a 2 ∈A a 1 , . . . , a n ∈A a 1 ·a n−1 , the terminal evaluation is a subset of the universe, F a 1 ·a n−1 (a n )⊆U. 23 Chapter 2. Types of Soft Set Remark 2.12.2.(i) Forn= 1, Definition 2.12.1 reduces to the usual (crisp) soft setF:A→ P(U). (i) Forn= 2,F:A→Σ (1) (U, E), i.e., each primary parametera∈Ais assigned a Type-1 soft set; this is the usual Type-2 soft set [75–78] viewpoint (parameterization over an already parameterized family). Example 2.12.3(Real-life example of a Type-3soft set: company selection by department→ criterion→strictness).LetUbe a set of job candidates: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . LetEbe a pool of evaluation parameters (used at all levels), including: E=e Alg , e Comm , e Exp , e Strong , e Moderate . Interprete Alg =“good algorithms”,e Comm =“good communication”,e Exp =“relevant experi- ence”, ande Strong , e Moderate asstrictness levels(meta-criteria) for acceptance. We construct a Type-3soft set(F, A)∈Σ (3) (U, E)that models the following hierarchy: Department−→Criterion−→Strictness. Level 1 (primary parameters): departments.Let A=a Eng , a PM ⊆E be the primary parameter set, wherea Eng =“Engineering department” anda PM =“Product management department”. For each departmenta∈A, we define a Type-2soft set F(a) = (F a , A a )∈Σ (2) (U, E). Level 2: criteria used by each department.Let A a Eng =e Alg , e Exp ⊆E,A a PM =e Comm , e Exp ⊆E. Thus Engineering focuses on(Alg,Exp), while PM focuses on(Comm,Exp). For each criterion c∈A a , we define a Type-1soft set F a (c) = (F a,c , A a,c )∈Σ (1) (U, E), whereA a,c is a strictness-parameter set andF a,c :A a,c → P(U)returns the candidate subset accepted under that strictness. Level 3: strictness parameters and final accepted subsets.For each(a, c)above, take A a,c =e Strong , e Moderate ⊆E. Define the terminal (Type-1) acceptance mappings as follows. 24 Chapter 2. Types of Soft Set Engineering→Algorithms: F a Eng ,e Alg (e Strong ) =u 1 , u 4 ,F a Eng ,e Alg (e Moderate ) =u 1 , u 3 , u 4 , u 6 . Engineering→Experience: F a Eng ,e Exp (e Strong ) =u 2 , u 4 ,F a Eng ,e Exp (e Moderate ) =u 1 , u 2 , u 4 , u 5 . PM→Communication: F a PM ,e Comm (e Strong ) =u 3 , u 5 ,F a PM ,e Comm (e Moderate ) =u 1 , u 3 , u 5 , u 6 . PM→Experience: F a PM ,e Exp (e Strong ) =u 2 , u 5 ,F a PM ,e Exp (e Moderate ) =u 2 , u 4 , u 5 , u 6 . Verification of the Type-3structure.By construction, for eacha∈A, the pair(F a , A a )is a Type-2soft set becauseF a :A a →Σ (1) (U, E). Moreover, for eachc∈A a , the pair(F a,c , A a,c ) is a Type-1soft set becauseF a,c :A a,c →P(U). Hence(F, A)∈Σ (3) (U, E)is a Type-3soft set. Interpretation.A chain(a, c, s)witha∈ Eng,PM,ca criterion used bya, ands∈ Strong,Moderateyields the final accepted subsetF a,c (s)⊆U. For example, the chain a Eng →e Alg →e Strong selects the candidatesu 1 , u 4 , whereas a PM →e Comm →e Moderate selectsu 1 , u 3 , u 5 , u 6 . 2.13 L-Soft Set AnL-soft set maps each parameter to anL-valued set on the universe, enabling lattice-graded, parameterized membership evaluations [79,80]. Definition 2.13.1(L-set).[79] LetXbe a nonempty universe and let(L,≤)be a bounded lattice (or, more generally, a poset of truth degrees with designated0 L ,1 L ). AnL-set(also called anL-fuzzy set) onXis a mapping A:X−→L. Forx∈X, the valueA(x)∈Lis interpreted as theL-valued grade of membership (truth degree) of the statement “x∈A”. We denote byL X the class of allL-sets onX. 25 Chapter 2. Types of Soft Set Definition 2.13.2(L-soft set).[79] LetXbe a nonempty universe, letEbe a nonempty set of parameters, and fix a bounded lattice(L,≤,0 L ,1 L ). LetA⊆Ebe nonempty. AnL-soft set overX(with parameter setA) is a pair Θ = (F, A), where F:A−→L X . Equivalently,Θcan be identified with a single mapping μ Θ :A×X−→L,μ Θ (a, x) :=F(a)(x), so that for each fixeda∈Athe sectionx7→μ Θ (a, x)is anL-set onX. Fora∈A, theL-set F(a)∈L X is called thea-approximation(ora-evaluation) ofΘ. Example 2.13.3(AnL-soft set for qualitative product-rating in e-commerce).LetXbe a finite set of smartphones X=p 1 , p 2 , p 3 , p 4 , and let the parameter set be A=Battery,Camera,Price. We use a bounded lattice of linguistic grades L=L,M,H,L≤M≤H,0 L =L,1 L =H, interpreted asLow,Medium, andHighsatisfaction, respectively. Define anL-soft setΘ = (F, A)by specifying, for each parametera∈A, anL-setF(a)∈L X (i.e., a mapX→L) that records the qualitative evaluation of each phone: F(Battery) :X→L,F(Battery)(p 1 ) =H, F(Battery)(p 2 ) =M, F(Battery)(p 3 ) =H, F(Battery)(p 4 ) =L, F(Camera) :X→L, F(Camera)(p 1 ) =M, F(Camera)(p 2 ) =H, F(Camera)(p 3 ) =M, F(Camera)(p 4 ) =H, F(Price) :X→L,F(Price)(p 1 ) =M, F(Price)(p 2 ) =L, F(Price)(p 3 ) =H, F(Price)(p 4 ) =M. Equivalently, the associated membership mapμ Θ :A×X→Lgiven byμ Θ (a, p) =F(a)(p) encodes, for each criteriona, the lattice-graded satisfaction level of each productp∈X. Interpretation.For example,μ Θ (Battery, p 3 ) =Hmeans thatp 3 is evaluated ashighon battery life, whileμ Θ (Price, p 2 ) =Lmeans thatp 2 is judgedlow(unfavorable) in price attractiveness. ThusΘmodels parameterized, qualitative (lattice-valued) assessments without using real-valued scores. 2.14 PosetSoft set (monotone soft set) A PosetSoft set is a soft set whose parameter order enforces monotonic inclusion: stronger parameters yield larger object subsets. 26 Chapter 2. Types of Soft Set Definition 2.14.1(PosetSoft set (monotone soft set)).LetUbe a nonempty universe and let (A,)be a nonempty partially ordered set of parameters. APosetSoft setoverUis a soft set (F, A)with F:A→P(U) satisfying the monotonicity constraint ab=⇒F(a)⊆F(b) (a, b∈A). Example 2.14.2(Example of a PosetSoft set: apartment shortlisting under increasing budget). Let the universe be a finite set of apartments U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . Assume their monthly rents (in thousand JPY) are: rent(u 1 ) = 75,rent(u 2 ) = 80,rent(u 3 ) = 92,rent(u 4 ) = 100,rent(u 5 ) = 108,rent(u 6 ) = 125. Let the parameter set be the set of budget thresholds A=80,100,120, equipped with the usual order:=≤(so80100120). Define a soft set(F, A)overUby F(b) =u∈U|rent(u)≤b(b∈A). Concretely, F(80) =u 1 , u 2 ,F(100) =u 1 , u 2 , u 3 , u 4 ,F(120) =u 1 , u 2 , u 3 , u 4 , u 5 . Then for anya, b∈A, ab=⇒F(a)⊆F(b), because increasing the allowable budget can only add (and never remove) feasible apartments. Hence(F, A)is a PosetSoft set overU. Theorem 2.14.3(Soft-set structure and well-definedness of PosetSoft sets).LetUbe a nonempty universe and let(A,)be a nonempty poset of parameters. LetF:A→ P(U)be a mapping satisfying ab=⇒F(a)⊆F(b) (a, b∈A). Then: (i)(Soft-set structure).(F, A)is a (classical) soft set overU. (i)(Well-defined monotonicity predicate).The monotonicity constraint is a well-defined property of(F, A), i.e., for each comparable pair(a, b)∈A×Awithab, the inclusion statementF(a)⊆F(b)is unambiguous. (i)(Induced order-preserving operator).Defineι:P(U) A → 0,1byι(F) = 1iffF satisfiesab⇒F(a)⊆F(b). Thenιis well-defined, and the class PosetSoft(U;A,) :=(F, A)|F:A→P(U), ι(F) = 1 of all PosetSoft sets overU(with parameter poset(A,)) is well-defined. 27 Chapter 2. Types of Soft Set (iv)(Canonical associated relation onU).Define a binary relation F onUby x F y:⇐⇒(∀a∈A) (x∈F(a)⇒y∈F(a)). Then F is well-defined and is a preorder onU(reflexive and transitive). Proof.(i) By assumptionA6 =∅andF:A→ P(U), hence(F, A)is a soft set overUby the standard definition. (i) For anya, b∈Awithab, the setsF(a)andF(b)are uniquely determined subsets of UbecauseFis a function. Therefore the statementF(a)⊆F(b)is unambiguous, and the implicationab⇒F(a)⊆F(b)is a well-defined predicate on(F, A). (i) The mapιassigns to each functionF∈ P(U) A a unique truth value in0,1, because the defining condition is a (well-defined) universal statement over the set of comparable pairs inA. Henceιis well-defined, and consequently the subset of soft sets satisfyingι(F) = 1is well-defined; this is exactlyPosetSoft(U;A,). (iv) We check reflexivity and transitivity. Reflexive.Fixx∈U. For everya∈A, the implicationx∈F(a)⇒x∈F(a)holds, hence x F x. Transitive.Assumex F yandy F z. Leta∈Aand supposex∈F(a). Fromx F ywe obtainy∈F(a), and then fromy F zwe obtainz∈F(a). Thus(∀a∈A)(x∈F(a)⇒z∈ F(a)), i.e.,x F z. Therefore F is a well-defined preorder onU. 2.15 Random soft set A random soft set is a measurable mapping from outcomes to soft sets, yielding parameter- indexed random subsets under uncertainty. Definition 2.15.1(Measurable space of soft sets).FixUandAas above and write S(U, A) := ( P(U) ) A =F:A→P(U), the set of all soft mappings on(U, A)(equivalently, all soft sets(F, A)overUwith this fixed parameter set). For eacha∈Aandu∈U, define themembership cylinder C a,u :=F∈S(U, A)|u∈F(a)⊆S(U, A). Let Σ U,A :=σ ( C a,u :a∈A, u∈U ) be theσ-algebra generated by all such cylinders. 28 Chapter 2. Types of Soft Set Definition 2.15.2(Random soft set).Let(Ω,F,P)be a probability space and fix a universe Uand parameter setA. Arandom soft set(on(U, A)) is an(F,Σ U,A )-measurable mapping F: (Ω,F)−→ ( S(U, A),Σ U,A ) . Forω∈Ω, writeF(ω) =F ω ∈S(U, A); then each outcomeωinduces a (deterministic) soft set (F ω , A)overU. Example 2.15.3(Random soft set: commuting routes under random traffic).LetU=r 1 , r 2 , r 3 , r 4 be a finite set of candidate commuting routes (e.g., train lines or driving routes), and let A=a Fast , a Cheap , a Safe be a parameter set, wherea Fast =“fast”,a Cheap =“cheap”, anda Safe =“low incident risk”. Model the (uncertain) morning condition by a finite probability space Ω =ω L , ω N , ω H ,F= 2 Ω ,P(ω L ) = 0.3,P(ω N ) = 0.5,P(ω H ) = 0.2, whereω L , ω N , ω H representlight/normal/heavycongestion (or disruption) states. For each outcomeω∈Ω, define a soft set(F ω , A)overUby specifyingF ω (a)⊆Uas the set of routes satisfying criterionaunder stateω. For instance, let: F ω (a Fast )F ω (a Cheap )F ω (a Safe ) ω=ω L r 1 , r 2 r 3 , r 4 r 2 , r 4 ω=ω N r 2 r 3 , r 4 r 2 , r 3 ω=ω H r 4 r 4 r 3 , r 4 (Interpretation: under heavy congestion, only router 4 remains “fast” and also “cheap”, while the “safe” set depends on disruption patterns.) Define the mapping F: Ω−→S(U, A),F(ω) := (F ω , A). SinceUandAare finite, the soft-set spaceSS(U, A)is finite; takingΣ U,A = 2 S(U,A) , the mapFis automatically(F,Σ U,A )-measurable. HenceFis arandom soft setin the sense of Definition 2.15.2: each realized traffic stateωinduces a deterministic soft set(F ω , A)describing, parameterwise, which routes are acceptable that day. Proposition 2.15.4(Pointwise measurability criterion).A mappingF: Ω→S(U, A)is a random soft set if and only if for everya∈Aandu∈Uthe event ω∈Ω :u∈F ω (a)∈F. Proof.By Definition 2.15.2,Fis measurable iffF −1 (B)∈ Ffor allB∈Σ U,A . SinceΣ U,A is generated by the cylindersC a,u (Definition 2.15.1), this is equivalent to requiringF −1 (C a,u )∈F for all(a, u). But F −1 (C a,u ) =ω∈Ω :F(ω)∈C a,u =ω∈Ω :u∈F ω (a), which yields the claim. 29 Chapter 2. Types of Soft Set Remark 2.15.5(Relation to random set theory).For each fixed parametera∈A, the coordi- nate map F a : Ω→P(U),F a (ω) :=F ω (a), is arandom subsetofUin the sense that all membership eventsω:u∈F a (ω)are measurable. Thus a random soft set is precisely afamily of random subsets indexed by parameters, packaged as a single measurable map into the soft-set space; this parallels the standard viewpoint of random sets as measurable set-valued mappings. 2.16 Capacitary soft set A capacitary soft set assigns each parameter a normalized monotone capacity onU, representing nonadditive, uncertainty-aware set evaluations. Definition 2.16.1(Capacitary soft set (nonadditive set-function valued)).LetUbe a nonempty finite universe and let Cap(U) := ν:P(U)→[0,1] ∣ ∣ ∣ ν(∅) = 0, ν(U) = 1, A⊆B⇒ν(A)≤ν(B) be the family of (normalized) capacities onU. LetAbe a nonempty parameter set. Acapacitary soft setoverUis a pair(F, A)with F:A→Cap(U). Example 2.16.2(Real-life example of a capacitary soft set: evaluating cybersecurity control bundles under different threat contexts).LetUbe a finite set of candidate cybersecurity controls: U=c 1 , c 2 , c 3 , wherec 1 =multi-factor authentication (MFA),c 2 =endpoint protection (EDR), andc 3 = network monitoring (NDR). LetA=a Low , a High be a parameter set of threat contexts (low- threat vs. high-threat season). A capacitary soft set(F, A)assigns to eacha∈Aa normalized capacityν a :P(U)→[0,1], which quantifies the (possibly nonadditive) overallrisk-reduction effectivenessof any bundle S⊆Uunder contexta. Defineν a Low by ν a Low (∅) = 0,ν a Low (U) = 1, and ν a Low (c 1 ) = 0.40,ν a Low (c 2 ) = 0.30,ν a Low (c 3 ) = 0.25, ν a Low (c 1 , c 2 ) = 0.70, ν a Low (c 1 , c 3 ) = 0.65, ν a Low (c 2 , c 3 ) = 0.55. Defineν a High by ν a High (∅) = 0,ν a High (U) = 1, and ν a High (c 1 ) = 0.45,ν a High (c 2 ) = 0.35,ν a High (c 3 ) = 0.35, ν a High (c 1 , c 2 ) = 0.85, ν a High (c 1 , c 3 ) = 0.80, ν a High (c 2 , c 3 ) = 0.75. 30 Chapter 2. Types of Soft Set Eachν a is a normalized capacity (monotone w.r.t.⊆, withν a (∅) = 0andν a (U) = 1), and it is typicallynonadditive; for instance, under high threat, ν a High (c 1 , c 2 ) = 0.85need not equalν a High (c 1 ) +ν a High (c 2 ) = 0.80, reflecting synergy/overlap effects among controls. Now defineF:A→Cap(U)by F(a Low ) =ν a Low ,F(a High ) =ν a High . Then(F, A)is a capacitary soft set overUin the sense of Definition 2.16.1. Theorem 2.16.3(Soft-set structure and well-definedness of capacitary soft sets).LetUbe a nonemptyfiniteuniverse, letAbe a nonempty parameter set, and let Cap(U) := ν:P(U)→[0,1] ∣ ∣ ∣ ν(∅) = 0, ν(U) = 1, X⊆Y⇒ν(X)≤ν(Y) . If(F, A)satisfiesF:A→Cap(U), then: (i)Soft-set structure.(F, A)is aT-valued soft set overUwith codomainT=Cap(U); equivalently, it is a mappingF:A→ T P(U) whose values are monotone set functions normalized byν(∅) = 0andν(U) = 1. (i)Well-defined parameter evaluations.For eacha∈A, the objectF(a)is a uniquely determined capacity onU, i.e., a uniquely determined function F(a) :P(U)→[0,1]withF(a)(∅) = 0, F(a)(U) = 1, X⊆Y⇒F(a)(X)≤F(a)(Y). (i)Well-defined induced evaluation map.The map Φ (F,A) :A×P(U)−→[0,1],Φ (F,A) (a, X) :=F(a)(X), is well-defined (single-valued). Proof.(i) By assumption,Ais nonempty andFis a mappingA→Cap(U). A (classical) soft set over a universeZwith parameter setAis any mapA→ P(Z). Here the “universe” being evaluated isP(U)and the codomain is notP(·)but the prescribed codomainT=Cap(U); thus(F, A)is precisely aT-valued soft set in the sense of Definition 2.19.1. Moreover, each F(a)∈Cap(U)is, by definition of Cap(U), a normalized monotone set function onP(U). (i) Fixa∈A. SinceFis a function, the valueF(a)is uniquely determined. Because the codomain ofFis Cap(U), we haveF(a)∈Cap(U), henceF(a) :P(U)→[0,1]and the three axiomsF(a)(∅) = 0,F(a)(U) = 1, andX⊆Y⇒F(a)(X)≤F(a)(Y)hold for allX, Y⊆U. Therefore each parameteradetermines a uniquely defined capacity. (i) DefineΦ (F,A) (a, X) :=F(a)(X)for(a, X)∈A× P(U). This is meaningful because, by (i), for each fixeda∈Athe expressionF(a)(X)is defined for everyX∈ P(U)and belongs to[0,1]. Uniqueness follows from the single-valuedness ofF(a)as a function. Hence Φ (F,A) :A×P(U)→[0,1]is well-defined. 31 Chapter 2. Types of Soft Set 2.17 CoverSoft set A CoverSoft set assigns each parameter a cover ofUby nonempty subsets, encoding parameter- dependent decompositions or granularizations. Definition 2.17.1(CoverSoft set).LetUbe a nonempty universe. Write Cov(U) := C ⊆P(U)\∅ ∣ ∣ ∣ ⋃ C∈C C=U for the family of all covers ofUby nonempty subsets. LetAbe a nonempty parameter set. A CoverSoft setoverUis a pair(F, A)with F:A→Cov(U). Example 2.17.2(Real-life example of a CoverSoft set: decomposing a delivery region into service zones under different strategies).LetUbe a finite universe of delivery addresses in a small region: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . LetA=a Geo , a Time be a set of operational parameters, wherea Geo denotes ageographic zoning strategyanda Time denotes atime-window zoning strategy. A CoverSoft set(F, A)assigns to eacha∈Aa coverF(a)∈Cov(U), i.e., a family of nonempty subsets ofUwhose union isU. (1) Geographic zones.Define F(a Geo ) = C 1 , C 2 , C 3 , where C 1 =u 1 , u 2 ,C 2 =u 3 , u 4 ,C 3 =u 5 , u 6 . Then eachC i 6 =∅and C 1 ∪C 2 ∪C 3 =U, soF(a Geo )∈Cov(U). (2) Time-window zones (overlapping cover).Define F(a Time ) = D 1 , D 2 , D 3 , where D 1 =u 1 , u 3 , u 5 (morning-feasible),D 2 =u 2 , u 3 , u 4 (afternoon-feasible),D 3 =u 4 , u 6 (evening-feasible). Again, eachD i 6 =∅and D 1 ∪D 2 ∪D 3 =u 1 , u 2 , u 3 , u 4 , u 5 , u 6 =U, henceF(a Time )∈Cov(U). (3) The CoverSoft set.Therefore the mappingF:A→Cov(U)given by F(a Geo ) =C 1 , C 2 , C 3 ,F(a Time ) =D 1 , D 2 , D 3 , defines a CoverSoft set(F, A)overUin the sense of Definition 2.17.1. 32 Chapter 2. Types of Soft Set Theorem 2.17.3(Soft-set structure and well-definedness of CoverSoft sets).LetUbe a nonempty universe and letAbe a nonempty parameter set. Define Cov(U) := C ⊆P(U)\∅ ∣ ∣ ∣ ⋃ C∈C C=U . Assume(F, A)satisfiesF:A→Cov(U). Then: (i)(Soft-set structure).(F, A)is aT-valued soft set overUin the sense of Definition 2.19.1 with codomainT=Cov(U). (i)(Well-defined parameterwise covers).For eacha∈A, the valueF(a)is a uniquely determined cover ofUby nonempty subsets; in particular, F(a)⊆P(U)\∅and ⋃ C∈F(a) C=U. (i)(Well-defined induced granulation neighborhoods).Define, for(a, u)∈A×U, N (F,A) (a, u) :=C∈F(a)|u∈C ⊆F(a). ThenN (F,A) :A×U→ P(P(U))is well-defined and satisfiesN (F,A) (a, u)6 =∅for all (a, u)∈A×U. Proof.First note that Cov(U)is nonempty becauseU∈Cov(U)(asU6=∅). (i) SinceA6 =∅andF:A→Cov(U), the pair(F, A)is exactly a soft set whose values lie in the fixed codomainT=Cov(U); this is precisely the notion of aT-valued soft set. (i) Fixa∈A. BecauseFis a function,F(a)is uniquely determined. Moreover,F(a)∈Cov(U) implies by definition thatF(a)⊆P(U)\∅and ⋃ C∈F(a) C=U. HenceF(a)is a well-defined cover ofUby nonempty subsets. (i) Fix(a, u)∈A×Uand defineN (F,A) (a, u) =C∈F(a) :u∈C. This set is well- defined because membershipu∈Cis unambiguous for eachC⊆U. To see nonemptiness, use ⋃ C∈F(a) C=Ufrom (i): sinceu∈U, there existsC∈F(a)withu∈C, henceN (F,A) (a, u)6 = ∅. ThereforeN (F,A) is a well-defined (everywhere nonempty) neighborhood/granulation map. 2.18 FiltrationSoft set A FiltrationSoft set assigns each parameter a nested chain of subsets, representing multi-level selection stages from strict to relaxed. 33 Chapter 2. Types of Soft Set Definition 2.18.1(FiltrationSoft set (multi-level subset output)).LetUbe a nonempty uni- verse and fix an integerk≥1. Define Fil k (U) := (S 0 , . . . , S k )∈P(U) k+1 ∣ ∣ ∣ S 0 ⊆S 1 ⊆·⊆S k . LetAbe a nonempty parameter set. AFiltrationSoft setoverU(of depthk) is a mapping F:A→Fil k (U). Example 2.18.2(Real-life example of a FiltrationSoft set: multi-stage hiring shortlists under different job profiles).LetUbe a finite set of job applicants: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 . Fix depthk= 3, so that each output is a nested chain (S 0 , S 1 , S 2 , S 3 )∈Fil 3 (U)withS 0 ⊆S 1 ⊆S 2 ⊆S 3 . Let the parameter set be A=a SE , a DS , wherea SE represents asoftware engineerprofile anda DS represents adata scientistprofile. DefineF:A→Fil 3 (U)by specifying, for each profilea∈A, four successive shortlists: •S 0 (a): candidates passing a strict initial screen (must-haves), •S 1 (a): candidates passing a technical screen, •S 2 (a): candidates passing interviews, •S 3 (a): candidates considered at all for that profile. (1) Software engineer profile.Let F(a SE ) = ( S SE 0 , S SE 1 , S SE 2 , S SE 3 ) := ( u 1 , u 2 ,u 1 , u 2 , u 4 ,u 1 , u 2 , u 4 , u 6 ,u 1 , u 2 , u 3 , u 4 , u 6 ) . Then S SE 0 ⊆S SE 1 ⊆S SE 2 ⊆S SE 3 ⊆U, soF(a SE )∈Fil 3 (U). (2) Data scientist profile.Let F(a DS ) = ( S DS 0 , S DS 1 , S DS 2 , S DS 3 ) := ( u 2 ,u 2 , u 5 ,u 2 , u 5 , u 7 ,u 2 , u 4 , u 5 , u 7 ) . Again, S DS 0 ⊆S DS 1 ⊆S DS 2 ⊆S DS 3 ⊆U, soF(a DS )∈Fil 3 (U). (3) FiltrationSoft set interpretation.HenceF:A→Fil 3 (U)is a FiltrationSoft set overU in the sense of Definition 2.18.1. Each parametera∈Aselects amulti-level(nested) sequence of acceptable applicants, representing progressively relaxed stages of the hiring pipeline for that job profile. 34 Chapter 2. Types of Soft Set Theorem 2.18.3(Soft-set structure and well-definedness of FiltrationSoft sets).LetUbe a nonempty universe, fix an integerk≥1, and define Fil k (U) := (S 0 , . . . , S k )∈P(U) k+1 ∣ ∣ ∣ S 0 ⊆S 1 ⊆·⊆S k . LetAbe a nonempty parameter set, and letF:A→Fil k (U)be a mapping. Then: (i)(Nonemptiness of the codomain).Fil k (U)6 =∅. (i)(Soft-set structure).(F, A)is aT-valued soft set overUin the sense of Definition 2.19.1 with codomainT=Fil k (U). (i)(Well-defined filtration at each parameter).For everya∈Athere exists a unique tuple F(a) = (S (a) 0 , . . . , S (a) k )∈P(U) k+1 satisfying the nesting chain S (a) 0 ⊆S (a) 1 ⊆·⊆S (a) k . (iv)(Well-defined stagewise soft sets).For eachi∈ 0,1, . . . , kdefine thei-th stage map F i :A−→P(U),F i (a) :=π i (F(a)), whereπ i :Fil k (U)→P(U)is the projectionπ i (S 0 , . . . , S k ) =S i . Then each(F i , A)is a classical soft set overU, and moreover the family is pointwise monotone: F i (a)⊆F i+1 (a) (∀a∈A,∀0≤i < k). Proof.(i) SinceU6=∅, the tuple(∅, . . . ,∅)∈ P(U) k+1 satisfies∅⊆ · ⊆∅, hence (∅, . . . ,∅)∈Fil k (U). Therefore Fil k (U)6=∅. (i) BecauseA6 =∅andF:A→Fil k (U), the pair(F, A)is, by definition, a soft set whose values lie in the fixed codomainT=Fil k (U); this is exactly aT-valued soft set as in Definition 2.19.1. (i) Fixa∈A. SinceFis a function,F(a)is uniquely determined. Also,F(a)∈Fil k (U)means precisely thatF(a)is a(k+ 1)-tuple of subsets ofU, sayF(a) = (S (a) 0 , . . . , S (a) k ), satisfying S (a) 0 ⊆·⊆S (a) k . Hence the filtration chain at parameterais well-defined. (iv) For each fixedi, the projectionπ i is a well-defined function from Fil k (U)toP(U), so F i =π i ◦F:A→ P(U)is well-defined, and therefore(F i , A)is a classical soft set overU. Finally, for anya∈A, the tupleF(a) = (S (a) 0 , . . . , S (a) k )lies in Fil k (U), soS (a) i ⊆S (a) i+1 for 0≤i < k. ButF i (a) =S (a) i andF i+1 (a) =S (a) i+1 by definition ofF i , henceF i (a)⊆F i+1 (a)for allaandi. 35 Chapter 2. Types of Soft Set 2.19T-valued soft set AT-valued soft set assigns each parameter aT-valued function on the universe, encoding pa- rameterized evaluations without requiring crisp subsets. Definition 2.19.1(T-valued soft set (general template)).LetXbe a nonempty universe, letE be a nonempty set of parameters, and letA⊆Ebe nonempty. LetTbe a nonempty codomain set and writeT X :=f:X→ T . AT-valued soft setoverX(with parameter setA) is a pair Θ = (F, A), where F:A−→T X . Equivalently,Θcan be identified with a single mapping μ Θ :A×X−→T,μ Θ (a, x) :=F(a)(x), so that for each fixeda∈Athe sectionx7→μ Θ (a, x)is aT-valued function onX. Theorem 2.19.2(Well-definedness and canonical identification ofT-valued soft sets).In the setting of Definition 2.19.1, the two descriptions F:A→T X andμ Θ :A×X→T, μ Θ (a, x) =F(a)(x), are equivalent in a canonical (bijection) sense. More precisely, the map Φ :T X A :=F:A→T X −→ T A×X :=μ:A×X→T,Φ(F)(a, x) :=F(a)(x), is a well-defined bijection with inverse Ψ :T A×X −→ T X A ,Ψ(μ)(a)(x) :=μ(a, x). Consequently, aT-valued soft setΘ = (F, A)determines a unique mapμ Θ :A×X→T, and conversely every suchμdetermines a uniqueT-valued soft set(Ψ(μ), A). Proof.Step 1 (Well-definedness ofΦ).LetF:A→ T X . For eacha∈A,F(a)∈ T X is (by definition ofT X ) a functionF(a) :X→ T. Hence for each(a, x)∈A×Xthe value F(a)(x)∈Tis uniquely determined. ThereforeΦ(F) :A×X→Tgiven by(a, x)7→F(a)(x) is a well-defined function. Step 2 (Well-definedness ofΨ).Letμ:A×X→T. Fixa∈A. DefineΨ(μ)(a) :X→T byx7→μ(a, x). This is a well-defined function inT X . Thusa7→Ψ(μ)(a)defines a well-defined mappingΨ(μ) :A→T X . Step 3 (ΦandΨare inverses).ForF:A→T X and any(a, x)∈A×X, (Φ◦Ψ)(Φ(F))(a, x) = Φ(F)(a, x) =F(a)(x), and more directly, (Ψ◦Φ)(F)(a)(x) = Φ(F)(a, x) =F(a)(x) (∀a∈A,∀x∈X), 36 Chapter 2. Types of Soft Set soΨ(Φ(F)) =Fas functionsA→T X . Similarly, forμ:A×X→Tand any(a, x)∈A×X, (Φ◦Ψ)(μ)(a, x) = Ψ(μ)(a)(x) =μ(a, x), soΦ(Ψ(μ)) =μas functionsA×X→T. ThusΦis bijective with inverseΨ. The stated uniqueness claims follow immediately from the existence of this bijection. Definition 2.19.3(Vector-valued soft set).LetXbe a nonempty universe, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. LetVbe a vector space over a fieldK. Avector- valued soft setoverX(with parameter setA) is aV-valued soft set Θ V = (F, A),F:A−→V X . Equivalently, it is a mapμ Θ V :A×X→Vgiven byμ Θ V (a, x) =F(a)(x). Example 2.19.4(Vector-valued soft set: multi-criteria scoring of job candidates).LetX= x 1 , x 2 , x 3 be three job candidates and letEbe a parameter pool. Choose the active parameter set A=Tech,Comm⊆E (technical skill and communication skill). LetV=R 2 . Define a vector-valued soft setΘ V = (F, A)by specifying, for eacha∈A, a functionF(a) : X→R 2 (soF:A→(R 2 ) X ): F(Tech)(x 1 ) = (0.80,0.70), F(Tech)(x 2 ) = (0.60,0.90), F(Tech)(x 3 ) = (0.75,0.65), F(Comm)(x 1 ) = (0.60,0.55), F(Comm)(x 2 ) = (0.40,0.60), F(Comm)(x 3 ) = (0.85,0.80). Interpretation: underTechthe two coordinates represent (coding, algorithms) scores, and under Commthey represent (presentation, teamwork) scores. Equivalently,μ Θ V :A×X→R 2 is given byμ Θ V (a, x) =F(a)(x). Definition 2.19.5(Matrix-valued soft set).LetXbe a nonempty universe, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. Fix integersm, n≥1and a fieldK, and denote by Mat m×n (K)the space ofm×nmatrices overK. Amatrix-valued soft setoverX(with parameter setA) is a Mat m×n (K)-valued soft set Θ M = (F, A),F:A−→Mat m×n (K) X . Equivalently, it is a mapμ Θ M :A×X→Mat m×n (K)withμ Θ M (a, x) =F(a)(x). Example 2.19.6(Matrix-valued soft set: shift-dependent state-transition estimates of ma- chines).LetX=M 1 , M 2 be two production machines and take A=Day,Night⊆E 37 Chapter 2. Types of Soft Set as operating-shift parameters. Fixm=n= 2andK=R, so the codomain is Mat 2×2 (R). Define a matrix-valued soft setΘ M = (F, A)by giving, for eacha∈A, a functionF(a) :X→ Mat 2×2 (R): F(Day)(M 1 ) = ( 0.97 0.03 0.40 0.60 ) ,F(Day)(M 2 ) = ( 0.95 0.05 0.30 0.70 ) , F(Night)(M 1 ) = ( 0.94 0.06 0.50 0.50 ) ,F(Night)(M 2 ) = ( 0.92 0.08 0.45 0.55 ) . Interpretation: each2×2matrix is a simple estimated transition matrix between statesOK,Fault for the corresponding shift (rows = current state, columns = next state). Equivalently,μ Θ M (a, x) = F(a)(x)∈Mat 2×2 (R). Definition 2.19.7(Tensor-valued soft set).LetXbe a nonempty universe, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. Fix a fieldKand vector spacesV 1 , . . . , V r over K(r≥1), and put T:=V 1 ⊗·⊗V r (ther-th order tensor space of type(V 1 , . . . , V r )). Atensor-valued soft setoverX(with param- eter setA) is aT-valued soft set Θ T = (F, A),F:A−→T X . Equivalently, it is a mapμ Θ T :A×X→Twithμ Θ T (a, x) =F(a)(x). Example 2.19.8(Tensor-valued soft set: store-dependent2×2×2demand-context tensor). LetX=S 1 , S 2 be two retail stores and let A=Weekday,Weekend⊆E be context parameters. FixK=Rand take V 1 =R 2 (demand: Low/High),V 2 =R 2 (weather: Cool/Hot),V 3 =R 2 (promotion: Off/On), soT=V 1 ⊗V 2 ⊗V 3 can be represented as2×2×2arrays. Define a tensor-valued soft setΘ T = (F, A)by specifyingF(a) :X→ T. For readability, we writeF(a)(x) = [t ijk ] i,j,k∈1,2 as two2×2slices (promotion Off/On): StoreS 1 . F(Weekday)(S 1 ) :k= 1 (Off)⇒ ( 0.30 0.10 0.20 0.05 ) , k= 2 (On)⇒ ( 0.15 0.05 0.10 0.05 ) , F(Weekend)(S 1 ) :k= 1 (Off)⇒ ( 0.20 0.10 0.25 0.10 ) , k= 2 (On)⇒ ( 0.10 0.05 0.15 0.05 ) . StoreS 2 . F(Weekday)(S 2 ) :k= 1 (Off)⇒ ( 0.35 0.10 0.15 0.05 ) , k= 2 (On)⇒ ( 0.20 0.05 0.08 0.02 ) . Interpretation:t ijk is a store- and context-dependent weight (e.g., empirical frequency) for the joint situation (demandi, weatherj, promotionk). Equivalently,μ Θ T (a, x) =F(a)(x)∈ V 1 ⊗V 2 ⊗V 3 . 38 Chapter 2. Types of Soft Set 2.20 Cubic Soft Set Cubic soft sets map each parameter to cubic sets: interval-valued membership degrees plus fuzzy membership, modeling uncertainty with dual information. The definition of the Cubic Soft Set is described as follows [81–83]. Definition 2.20.1(Cubic Soft Set).[81] LetXbe a nonempty universe and letEbe a set of parameters. First, acubic setinXis defined as a mapping that assigns to each elementx∈X a pair 〈[A − (x), A + (x)], λ(x)〉, where: •[A − (x), A + (x)]⊆[0,1]is an interval representing the degree of membership as given by aninterval-valued fuzzy set, and •λ(x)∈[0,1]is the membership degree provided by afuzzy set. Then, acubic soft setoverXwith respect to the parameter setEis a mapping F:E→cubic sets inX. Equivalently, a cubic soft set can be expressed as the collection ̃ F=(e, ̃ F(e)) :e∈E, where for each parametere∈E, the set ̃ F(e)is a cubic set inX; that is, ̃ F(e) =〈x,[A − e (x), A + e (x)], λ e (x)〉:x∈X. Example 2.20.2(Real-life example of a cubic soft set: apartment screening with uncertain scores and point estimates).LetXbe a set of apartments: X=a 1 , a 2 , a 3 , a 4 . LetEbe a set of decision parameters and consider E=e Safe , e Comm , wheree Safe =“neighborhood safety” ande Comm =“commute convenience”. A cubic soft setF:E→ cubic sets inXassigns to each parametere∈Ea cubic set onX of the form ̃ F(e) = 〈 x,[A − e (x), A + e (x)], λ e (x) 〉 ∣ ∣ ∣ x∈X , where[A − e (x), A + e (x)]is an interval-valued membership (reflecting uncertainty in the score) and λ e (x)is a single membership degree (a point estimate). For instance, based on crime statistics and expert judgment, suppose safety is assessed as: ̃ F(e Safe ) = 〈a 1 ,[0.70,0.85],0.80〉,〈a 2 ,[0.40,0.60],0.50〉,〈a 3 ,[0.55,0.75],0.65〉,〈a 4 ,[0.20,0.35],0.30〉 . Similarly, using travel-time variability data, suppose commute convenience is assessed as: ̃ F(e Comm ) = 〈a 1 ,[0.30,0.50],0.40〉,〈a 2 ,[0.75,0.90],0.85〉,〈a 3 ,[0.50,0.70],0.60〉,〈a 4 ,[0.60,0.80],0.70〉 . ThenFis a cubic soft set onX: each parametereis associated with a cubic evaluation of every apartment, combining an interval-valued degree (uncertainty band) with a representative point degree. In decision-making, one may aggregate ̃ F(e Safe )and ̃ F(e Comm )to rank apartments under both safety and commuting considerations. 39 Chapter 2. Types of Soft Set 2.21 Probabilistic Soft Set A probabilistic soft set maps each parameter to a probability distribution over the universe, modeling uncertainty via normalized likelihoods [84,85]. Definition 2.21.1(Probabilistic Soft Set).[84,85] LetUbe a non-empty finite universe and Ebe a set of parameters. LetA⊆Ebe a subset of parameters. Denote by D(U) = μ:U→[0,1] ∣ ∣ ∣ ∑ u∈U μ(u) = 1 the set of all probability distributions onU. Aprobabilistic soft setoverUis a pair(F, A) where F:A→D(U) such that for eache∈A, the functionF(e) :U→[0,1]satisfies ∑ u∈U F(e)(u) = 1. In other words, for every parametere∈A,F(e)is a probability distribution onU. Example 2.21.2(Real-life example of a probabilistic soft set: choosing a commuting route under different criteria).LetUbe a finite set of candidate commuting routes: U=r 1 , r 2 , r 3 , r 4 . LetEbe a set of decision parameters and take A=e Fast , e Cheap , e Comfort ⊆E, wheree Fast =“fastest”,e Cheap =“cheapest”, ande Comfort =“most comfortable”. A probabilistic soft set(F, A)assigns to each parametere∈Aa probability distribution onU, interpreted as the likelihood that each route is the best choice under criterione(e.g., estimated from historical travel data and user preference models). For instance, defineF:A→D(U)by the following distributions: F(e)(u)r 1 r 2 r 3 r 4 F(e Fast )(·)0.55 0.25 0.15 0.05 F(e Cheap )(·)0.10 0.20 0.60 0.10 F(e Comfort )(·)0.15 0.50 0.10 0.25 Each row sums to1, so eachF(e)is a probability distribution onU. Thus(F, A)is a probabilistic soft set overU. Interpretation: under “fastest” the model favors router 1 , under “cheapest” it favorsr 3 , and under “most comfortable” it favorsr 2 . The comparison of classical soft sets and probabilistic soft sets is presented in Table 2.4. 40 Chapter 2. Types of Soft Set Table 2.4: Concise comparison of classical soft sets and probabilistic soft sets over a finite universe U. AspectSoft setProbabilistic soft set Universe/parametersFinite (or general) universeU and parameter setA⊆E. Finite universeUand parame- ter setA⊆E(typically finite to interpret distributions). Value assigned to a parametera∈A crisp subsetF(a)⊆U.Aprobabilitydistribu- tionF(a)∈D(U), i.e. F(a) :U→[0,1]with ∑ u∈U F(a)(u) = 1. Mathematical type ofFF:A→P(U).F:A→D(U)⊆[0,1] U . Interpretation“Accepted/feasible objects un- der parametera” (yes/no mem- bership). “Likelihood/degree of prefer- ence of each object under pa- rametera” (normalized uncer- tainty). Aggregation across parametersOften via set operations (union/intersection) or count- ing scores. Often via probabilistic com- bination (e.g.,weighted mixtures, Bayesian updates, expected-utility rules). Relation between the twoBaseline model with crisp infor- mation. Generalizes soft sets: a soft set can be embedded by us- ing point-mass distributions on F(a)(or thresholdingF(a)). 2.22 D-soft set A D-soft set maps each parameter to a D-number mass assignment on subsets, handling incom- pleteness and nonexclusive evidence. Definition 2.22.1(D-number onU).[86,87] LetUbe a nonemptyfiniteuniverse. AD-number onUis a mapping D: 2 U −→[0,1] satisfying D(∅) = 0, ∑ B⊆U D(B)≤1. (Unlike classical Dempster–Shafer basic probability assignments, D-number theory does not re- quire the elements ofUto be mutually exclusive, and it allows incomplete information when the above sum is<1.) Let DNum(U) := D: 2 U →[0,1] ∣ ∣ ∣ D(∅) = 0, ∑ B⊆U D(B)≤1 be the set of all D-numbers onU. Definition 2.22.2(D-soft set).LetUbe a nonemptyfiniteuniverse and letEbe a nonempty set of parameters. LetA⊆Ebe nonempty. AD-soft setoverUwith parameter setAis a pair (F, A)where F:A−→DNum(U). Thus, for each parametere∈A, the valueF(e) =D e is a D-number onU, i.e., a mass assignment D e : 2 U →[0,1]with ∑ B⊆U D e (B)≤1. 41 Chapter 2. Types of Soft Set Example 2.22.3(Real-life example of a D-soft set: supplier selection under incomplete and nonexclusive evidence).LetUbe a finite set of candidate suppliers for a manufacturing order: U=s 1 , s 2 , s 3 , s 4 . LetEbe a set of evaluation parameters and take the nonempty subset A=e Qual , e Deliv ⊆E, wheree Qual =“high product quality” ande Deliv =“reliable delivery”. A D-soft set(F, A)assigns to each parametere∈Aa D-numberD e ∈DNum(U), i.e., a map D e : 2 U →[0,1]withD e (∅) = 0and ∑ B⊆U D e (B)≤1. (1) Evidence for quality.Suppose quality audits provide the following (possibly incomplete) evidence: D e Qual (s 1 ) = 0.45,D e Qual (s 1 , s 3 ) = 0.20,D e Qual (s 2 ) = 0.10, D e Qual (s 2 , s 4 ) = 0.05,D e Qual (B) = 0for all otherB⊆U. Then ∑ B⊆U D e Qual (B) = 0.45 + 0.20 + 0.10 + 0.05 = 0.80≤1, soD e Qual ∈DNum(U); the remaining mass1−0.80 = 0.20represents unassigned (unknown) information. (2) Evidence for delivery.From shipping records and logistics reports, suppose we obtain: D e Deliv (s 3 ) = 0.30,D e Deliv (s 1 , s 3 ) = 0.25,D e Deliv (s 1 , s 2 , s 3 ) = 0.10, D e Deliv (B) = 0for all otherB⊆U. Hence ∑ B⊆U D e Deliv (B) = 0.30 + 0.25 + 0.10 = 0.65≤1, soD e Deliv ∈DNum(U). Note that the focal setss 3 ⊆s 1 , s 3 ⊆s 1 , s 2 , s 3 arenotmutually exclusive, which is allowed in D-number modeling. (3) The D-soft set.DefineF:A→DNum(U)by F(e Qual ) =D e Qual ,F(e Deliv ) =D e Deliv . Then(F, A)is a D-soft set overU: each parameter is associated with a D-number encoding subset-valued evidence, while permitting incompleteness ( ∑ D e <1) and nonexclusive focal sets. A comparison between a classical soft set and a D-soft set is presented in Table 2.5. 42 Chapter 2. Types of Soft Set Table 2.5: Concise comparison between a classical soft set and a D-soft set over a (finite) universe U. AspectClassical soft set(F, A)D-soft set(F, A) Universe requirementUany nonempty set (often finite in applications). Uis assumed finite since each value is a D-number on2 U . Parameter domainNonemptyA⊆E.NonemptyA⊆E. Codomain / value typeF(e)∈ P(U)(a crisp subset of alternatives). F(e) =D e ∈DNum(U)where D e : 2 U →[0,1](a subset-mass assignment). Meaning ofF(e)Objectsacceptedunder parameter e(yes/no selection). Evidence aboutwhich subset(s)of Uare plausible undere, via focal sets with masses. Granularity of informationElement-level inclusion only.Subset-level evidence; can express ambiguity such as “eitheru 1 or u 3 ” by mass onu 1 , u 3 . Normalization / completeness No probability-like normalization constraint. ∑ B⊆U D e (B)≤1; the gap 1− ∑ B D e (B)represents unas- signed/unknown information. Mutual exclusivity requirement Not an evidence model; typically treats alternatives crisply. D-number framework does not re- quire mutual exclusivity of focal sets (and may model nonexclusive evidence). Typical decision extractionScoring/ranking by counting sat- isfied parameters (or other set- based aggregations). Ranking via belief/plausibil- ity/credibility transforms or other mass-to-score rules (application- dependent). 2.23 Complex Soft Sets A complex soft set assigns to each parameter a complex-valued membership function on the universe, thereby encoding both magnitude and phase information. Related notions include complex fuzzy sets[88,89] andcomplex neutrosophic sets[90–92]. Definition 2.23.1(Complex fuzzy set).[93, 94] LetUbe a nonempty universe. Define the closed unit disk D:=z∈C||z|≤1. Acomplex fuzzy set(CFS) onUis a mapping μ A :U−→D. Equivalently, for eachu∈Uwe may write μ A (u) =r A (u)e iω A (u) ,r A (u)∈[0,1], ω A (u)∈[0,2π], wherer A (u)is theamplitude(membership magnitude) andω A (u)is thephase. We denote by CFS(U)the family of all complex fuzzy sets onU. Definition 2.23.2(Complex soft set).LetUbe a nonempty universe and letEbe a nonempty set of parameters. LetA⊆Ebe nonempty. Acomplex soft set(CSS) overUwith parameter setAis a pair(F, A)where F:A−→CFS(U). 43 Chapter 2. Types of Soft Set Thus, for each parametere∈A, the valueF(e)is a complex fuzzy set onU, i.e., F(e) =μ e :U−→D. Equivalently, a complex soft set can be identified with a single mapping μ:A×U−→D,μ(e, u) :=μ e (u), such that for every fixede∈A, the sectionμ e (·)is a complex fuzzy membership function onU. Remark 2.23.3(Reductions).1. Ifμ(e, u)∈[0,1]⊂C(equivalently, all phases are0), then (F, A)reduces to a fuzzy soft set. 2. Ifμ(e, u)∈0,1for all(e, u), then(F, A)reduces to a (crisp) soft set. Example 2.23.4(Real-life example of a complex soft set: wearable-sensor sleep assessment with confidence phase).LetUbe a set of nights (sleep records) for a person: U=n 1 , n 2 , n 3 , n 4 . LetEbe a set of assessment parameters and take A=e Deep , e Stress ⊆E, wheree Deep =“deep-sleep quality” ande Stress =“low night-time stress”. A complex soft set(F, A)assigns to each parametere∈Aa complex fuzzy setμ e :U→D= z∈C||z|≤1. Interpret theamplituder(e, n)∈[0,1]as thedegreeto which nightnsatisfies e, and interpret thephaseω(e, n)∈[0,2π]as aconfidence/regularity markerderived from signal quality (e.g., stable sensors yield small phase, noisy sensors yield larger phase). Defineμ:A×U→Dbyμ(e, n) =r(e, n)e iω(e,n) with the following values: μ(e Deep , n 1 ) = 0.80e i0.10π , μ(e Deep , n 2 ) = 0.45e i0.35π , μ(e Deep , n 3 ) = 0.70e i0.15π , μ(e Deep , n 4 ) = 0.20e i0.60π , μ(e Stress , n 1 ) = 0.60e i0.20π , μ(e Stress , n 2 ) = 0.30e i0.55π , μ(e Stress , n 3 ) = 0.75e i0.10π , μ(e Stress , n 4 ) = 0.50e i0.40π . For each fixede∈A, the sectionμ e (·) =μ(e,·) :U→Dis a complex membership function, so it defines a complex fuzzy setF(e) =μ e ∈CFS(U). Hence(F, A)(equivalentlyμ) is a complex soft set. Interpretation: the magnitude|μ(e, n)|=r(e, n)expresses how well the nightnsatisfies the criterione, while the phase arg(μ(e, n)) =ω(e, n)encodes a secondary aspect such as confidence or signal stability, which is useful when sensor quality varies across nights. 44 Chapter 2. Types of Soft Set 2.24 Real Soft Set A real soft set maps each parameter to a bounded nonempty subset of real numbers, representing parameter-dependent possible real values [95–97]. Definition 2.24.1(Soft real set (real soft set)).[97] LetAbe a nonempty set of parameters and letRbe the set of real numbers. Define the collection of all nonempty bounded subsets of Rby P b (R) :=B⊆R|B6 =∅andBis bounded. Asoft real set(also called areal soft set) overRwith parameter setAis a pair(F, A), where F:A−→P b (R) is a mapping. For eachλ∈A, the valueF(λ)⊆Ris interpreted as the (parameter-dependent) set of possible real values under the parameterλ. Remark 2.24.2(Soft real numbers as singleton soft real sets).If(F, A)is asingletonsoft real set, i.e.,F(λ) =r(λ)for everyλ∈A, then it is naturally identified with the corresponding soft real number ̃r:A−→R, ̃r(λ) =r(λ). Example 2.24.3(Real-life example of a soft real set: delivery-time windows under different shipping options).LetAbe a set of shipping options for an online store: A=λ Std , λ Exp , λ Eco , whereλ Std = standard shipping,λ Exp = express shipping, andλ Eco = economy shipping. Consider the real quantity “delivery time” measured in days, so the value domain isR. Because delivery times are uncertain but bounded for each option, define F:A−→P b (R) by assigning to each option the (bounded, nonempty) set of plausible delivery times: F(λ Std ) = [2,5],F(λ Exp ) = [1,2],F(λ Eco ) = [4,9]. EachF(λ)⊆Ris nonempty and bounded, henceF(λ)∈P b (R). Therefore(F, A)is a soft real set overR. Interpretation: the same order may have different feasible delivery-time windows depending on the chosen shipping parameterλ∈A. 45 Chapter 2. Types of Soft Set 2.25 Intersectional soft sets An intersectional soft set satisfiesF(x)∩F(y)⊆F(x∗y), ensuring combined parameters include all jointly satisfying objects [98–101]. Definition 2.25.1(Intersectional soft set).[98–101] LetUbe a universal set and letEbe a set of parameters equipped with a binary operation ∗:E×E−→E. LetA⊆Ebe a nonempty subset and let(F, A)be a soft set overU, i.e., F:A−→P(U). Then(F, A)is called anintersectional soft setoverU(with respect to∗) if, for allx, y∈Asuch thatx∗y∈A, one has F(x)∩F(y)⊆F(x∗y). Remark 2.25.2.The condition says that whenever the “combined” parameterx∗yis admissible (lies inA), it must contain all objects that satisfy bothxandysimultaneously. Example 2.25.3(Real-life example of an intersectional soft set: product filtering with a “bun- dle” operation).LetUbe a set of products in an online store: U=p 1 , p 2 , p 3 , p 4 , p 5 , p 6 . LetEbe a set of product-tag parameters and take A=e Bio , e GF , e BioGF ⊆E, where e Bio =“organic”,e GF =“gluten-free”,e BioGF =“organic and gluten-free”. Define a binary operation∗:E×E→E(a “bundle” of tags) on the relevant parameters by e Bio ∗e GF =e BioGF ,e GF ∗e Bio =e BioGF , and setx∗x=xforx∈A(idempotence onA). Define a soft mappingF:A→P(U)by listing products that satisfy each tag: F(e Bio ) =p 1 , p 2 , p 4 ,F(e GF ) =p 2 , p 3 , p 5 ,F(e BioGF ) =p 2 . Then F(e Bio )∩F(e GF ) =p 2 ⊆F(e BioGF ), so the intersectional conditionF(x)∩F(y)⊆F(x∗y)holds forx=e Bio andy=e GF (and similarly for the symmetric order). Interpretation: the combined tag parametere BioGF must include every product that is both organic and gluten-free, ensuring consistency of the tag-bundling rule. 46 Chapter 2. Types of Soft Set 2.26N-soft Sets AnN-soft set assigns to each object, for each parameter, a discrete grade from0, . . . , N−1, thereby yielding graded approximations [102–105]. Related notions includeN-HyperSoft sets [106,107]. Definition 2.26.1(N-soft set).[102,103] LetUbe a nonempty universe of discourse and let Ebe a nonempty set of parameters. Fix an integerN≥2and let G N :=0,1, . . . , N−1 be a set of (ordered) grades. LetA⊆Ebe nonempty. A triple(F, A, N)is called anN-soft setonU(with parameter setA) if F:A−→P(U×G N ) satisfies theuniqueness-of-gradecondition: for everya∈Aand everyu∈U, there exists a uniquer∈G N such that (u, r)∈F(a). Equivalently, for eacha∈Athe setF(a)⊆U×G N is the graph of a unique function g a :U−→G N ,g a (u) =r⇐⇒(u, r)∈F(a), and hence the wholeN-soft set can be identified with a singleinformation function g:A×U−→G N ,g(a, u) :=g a (u). Example 2.26.2(Real-life example of anN-soft set: grading students by discrete performance levels).LetUbe a set of students in a class: U=s 1 , s 2 , s 3 , s 4 , s 5 . LetEbe a set of evaluation parameters and take A=a Math , a Prog ⊆E, wherea Math =“mathematics” anda Prog =“programming”. FixN= 5and hence G 5 =0,1,2,3,4, interpreted as ordered grades 0 =very poor≺1 =poor≺2 =average≺3 =good≺4 =excellent. Define the information functiong:A×U→G 5 by discrete rubric-based assessments: g(a, u) a Math a Prog s 1 43 s 2 24 s 3 12 s 4 31 s 5 02 47 Chapter 2. Types of Soft Set Equivalently, defineF:A→P(U×G 5 )by taking graphs of the grade functionsg a (u) :=g(a, u): F(a Math ) =(s 1 ,4),(s 2 ,2),(s 3 ,1),(s 4 ,3),(s 5 ,0), F(a Prog ) =(s 1 ,3),(s 2 ,4),(s 3 ,2),(s 4 ,1),(s 5 ,2). For eacha∈Aandu∈U, there is a uniquer∈G 5 such that(u, r)∈F(a), so(F, A,5)is an N-soft set onU. Interpretation: each student receives adiscretegrade for each parameter (subject), and the resultingN-soft set supports graded decision rules such as selecting students withg(a Prog , u)≥3. 2.27n-ary soft set Ann-ary soft set maps each parameter to ann-tuple of subsets over multiple universes, enabling multi-attribute approximations. Definition 2.27.1(Binary soft set).[108,109] LetU 1 andU 2 be two nonempty universe sets, and letEbe a set of parameters. Fix a nonempty parameter subsetA⊆E. Abinary soft set over(U 1 , U 2 )(with parameter setA) is a pair(F, A), where F:A−→P(U 1 )×P(U 2 ). For eache∈A, we write F(e) = (X e , Y e ),X e ⊆U 1 , Y e ⊆U 2 . Equivalently,(F, A)can be identified with two ordinary soft sets(F 1 , A)overU 1 and(F 2 , A) overU 2 , whereF 1 (e) :=X e andF 2 (e) :=Y e for alle∈A. Definition 2.27.2(n-ary soft set).Letn∈Nwithn≥2, letU 1 , . . . , U n be nonempty universe sets, and letEbe a set of parameters. Fix a nonempty parameter subsetA⊆E. Ann-ary soft setover(U 1 , . . . , U n )(with parameter setA) is a pair(F, A), where F:A−→ n ∏ i=1 P(U i ). Thus, for each parametere∈Awe have ann-tuple F(e) = (X (1) e , X (2) e , . . . , X (n) e ),X (i) e ⊆U i (i= 1, . . . , n). Equivalently,(F, A)is uniquely determined by ann-tuple of (ordinary) soft sets (F i , A)overU i (i= 1, . . . , n), via the component mapsF i :A→P(U i )defined by F i (e) :=X (i) e (e∈A, i= 1, . . . , n), so thatF(e) = (F 1 (e), . . . , F n (e))for alle∈A. 48 Chapter 2. Types of Soft Set 2.28 Linguistic Soft Set A linguistic soft set maps parameters to subsets of the universe by using ordered linguistic terms, thereby enabling qualitative evaluations instead of numeric membership degrees. Linguistic hypersoft sets have been studied in the literature [110, 111]; here we rewrite the concept in the (classical)linguistic soft setform. Related models have also been investigated in various contexts [112–116]. Definition 2.28.1(Linguistic Soft Set).[110, 111] LetΩbe a finite universe of objects (e.g., rural health service centers), and letEbe a nonempty set of parameters (criteria). Fix a nonempty subsetA⊆Eof parameters to be used. For each parametere∈A, letΥ e be a finite, strictly ordered set of linguistic values, Υ e =κ e,1 , κ e,2 , . . . , κ e,m e ,κ e,1 ≺κ e,2 ≺·≺κ e,m e , (where, for example,κ e,1 =“very low” andκ e,m e =“very high”). Let Υ := ⊔ e∈A Υ e denote the disjoint union of the linguistic term sets. Alinguistic soft set(LSS) overΩwith parameter setAis a pair(Γ, A), where Γ :A−→P(Ω)×Υ is a mapping such that, for eache∈A, there exists a linguistic termκ e ∈Υ e with Γ(e) = ( Γ e , κ e ) ,Γ e ∈P(Ω), κ e ∈Υ e . Equivalently, one may view an LSS as an annotated soft set (Γ, A) = ( e, κ e ,Γ e ) ∣ ∣ ∣ e∈A , whereΓ e is the subset of objects described (or selected) by the linguistic evaluationκ e under the criterione. In applications, the annotationκ e is determined by experts or data-driven rules, andΓ e collects the objects inΩthat satisfy the parametereat the linguistic levelκ e . Example 2.28.2(Real-life example of a Linguistic Soft Set: rating restaurants by linguistic service/price levels).LetΩbe a small set of restaurants: Ω =r 1 , r 2 , r 3 , r 4 , r 5 . LetEbe a set of evaluation criteria and take A=e Srv , e Price ⊆E, wheree Srv =“service quality” ande Price =“price level”. 49 Chapter 2. Types of Soft Set For eache∈A, fix an ordered set of linguistic values: Υ e Srv =poor≺fair≺good≺excellent,Υ e Price =cheap≺moderate≺expensive. LetΥ = Υ e Srv tΥ e Price . DefineΓ :A→ P(Ω)×Υby assigning to each criterion a linguistic label together with the subset of restaurants that match that label (according to aggregated reviews): Γ(e Srv ) = ( r 1 , r 3 ,excellent ) ,Γ(e Price ) = ( r 2 , r 5 ,cheap ) . Equivalently, the linguistic soft set is the annotated collection (Γ, A) = ( e Srv ,excellent,r 1 , r 3 ) , ( e Price ,cheap,r 2 , r 5 ) . Interpretation: under the criterion “service quality”, restaurantsr 1 andr 3 are assessed asex- cellent; under “price level”, restaurantsr 2 andr 5 are assessed ascheap. Such a linguistic soft set supports qualitative filtering (e.g., seeking restaurants with excellent service or cheap price) without introducing numeric membership degrees. 2.29 MetaSoft Set A MetaSoft set is a soft set whose universe consists of soft sets, classifying families of soft sets by meta-parameters [117]. We first fix a general single-sorted, finitarysignature Σ = ( Func,Rel,ar Func ,ar Rel ) , whereFunc(resp.Rel) is a set of function (resp. relation) symbols, and ar records arities. A (single-sorted)Σ-structureis C= ( H,(f C ) f∈Func ,(R C ) R∈Rel ) , with carrierH6 =∅, interpretationsf C :H m →Hfor eachf∈Funcof aritym, and relations R C ⊆H r for eachR∈Relof arityr. Let Str Σ denote the class of allΣ-structures. Definition 2.29.1(MetaStructure over a fixed signature).(cf. [118]) FixΣas above. AMetaS- tructure(“structure of structures”) overΣis a pair M= ( U,(Φ ` ) `∈Λ ) , where: •Uis a nonempty set withU⊆Str Σ (its elements areobjectsat level 0); • for each label`∈Λofmeta-arityk ` ∈N, themeta-operation Φ ` :U k ` −→U 50 Chapter 2. Types of Soft Set is specified by uniformcarrier- and symbol-constructors: Γ ` : (C 1 , . . . ,C k ` )7→H ` (new carrierH ` built functorially); ∀f∈Func: f Φ ` (C 1 ,...,C k ` ) = Λ f ` ( f C 1 , . . . , f C k ` ) ; ∀R∈Rel: R Φ ` (C 1 ,...,C k ` ) = Ξ R ` ( R C 1 , . . . , R C k ` ) , whereΛ f ` andΞ R ` areuniformrecipes turning the symbols’ interpretations on inputs into the symbol’s interpretation on the output, over the new carrierH ` . Moreover, eachΦ ` isisomorphism-invariant(a.k.a. natural): ifα i :C i ∼ = D i for1≤i≤k ` , then there is an induced isomorphism Φ ` (α 1 , . . . , α k ` ) : Φ ` (C 1 , . . . ,C k ` ) ∼ = −→Φ ` (D 1 , . . . ,D k ` ) commuting with all interpretations of symbols ofΣ. Definition 2.29.2(MetaSoft Set).LetUbe a nonempty universe of objects and letSbe a (possibly finite) set of parameters. A (crisp) soft set on(U, S)is a mapping F:S−→P(U), and we denote by Soft(U, S) := F |F:S→P(U) the collection of all such soft sets on(U, S). LetΠbe a nonempty set ofmeta-parameters. AMetaSoft Set on(U, S)with meta-parameter setΠis a soft set over the universeSoft(U, S)with parameter setΠ, that is, a pair (G,Π)where G: Π−→P ( Soft(U, S) ) . For eachπ∈Π, the valueG(π)⊆Soft(U, S)is interpreted as thefamily of base soft setsthat satisfy the meta-criterion encoded byπ. Example 2.29.3(Real-life example of a MetaSoft Set: selecting suitable recommendation pro- files).LetUbe a set of customers of an online grocery service: U=c 1 , c 2 , c 3 , c 4 . LetSbe a set of product-category parameters: S=s V , s G , s L , wheres V = “prefers vegan items”,s G = “prefers gluten-free items”,s L = “prefers low-sugar items”. 51 Chapter 2. Types of Soft Set A (base) soft set on(U, S)is a mapF:S→P(U). Consider three candidate recommendation profiles (three base soft sets)F 1 ,F 2 ,F 3 ∈Soft(U, S)defined by: F 1 (s V ) =c 1 , c 3 ,F 1 (s G ) =c 2 , c 3 ,F 1 (s L ) =c 1 , c 2 , F 2 (s V ) =c 1 , c 2 , c 3 ,F 2 (s G ) =c 3 ,F 2 (s L ) =c 1 , F 3 (s V ) =c 4 ,F 3 (s G ) =c 2 , c 4 ,F 3 (s L ) =c 2 , c 3 , c 4 . ThusF 1 ,F 2 ,F 3 ⊆Soft(U, S)is a small universe of possible “recommendation rules” (soft-set profiles). Now letΠbe a set ofmeta-parametersdescribing constraints on such profiles: Π =π Bal , π Inc , where π Bal =“balanced coverage”,π Inc =“inclusive vegan”. DefineG: Π→P(Soft(U, S))by G(π Bal ) = F ∈Soft(U, S) ∣ ∣ ∣ ∣ ∣ F(s V ) ∣ ∣ ≥2, ∣ ∣ F(s G ) ∣ ∣ ≥2, ∣ ∣ F(s L ) ∣ ∣ ≥2 , G(π Inc ) = F ∈Soft(U, S) ∣ ∣ ∣ F(s V )⊇c 1 , c 3 . Then(G,Π)is a MetaSoft Set: for each meta-parameterπ∈Π,G(π)is a family ofbase soft sets(recommendation profiles) satisfying the meta-criterionπ. For the concrete profiles above, one checks that F 1 ∈G(π Bal ),F 2 ∈G(π Inc ),F 3 /∈G(π Inc ), illustrating how a MetaSoft Set can be used to select or filter candidate soft-set profiles according to higher-level design requirements. 2.30 Double-framed Soft Set A double-framed soft set assigns each parameter positive and negative approximation subsets, often constrained by an operation on parameters [119–122]. Moreover, further extensions have been studied, includingN-framed soft sets [123–125], double-framed HyperSoft sets [126, 127], and Double-Framed SuperHyperSoft Set [128,129]. Definition 2.30.1(Double-Framed Soft Set).LetUbe a universal set and letAbe a (nonempty) set of parameters. Assume thatAis equipped with a binary operation ∗:A×A−→A. Adouble-framed soft setoverU(with parameter setA) is a triple 〈(α, β);A〉, where α:A→P(U)andβ:A→P(U) are mappings. For eachx∈A, the setα(x)is interpreted as thepositive frameandβ(x)as the negative frame. Moreover, the following compatibility conditions are required: for allx, y∈A, α(x∗y)⊇α(x)∩α(y),β(x∗y)⊆β(x)∪β(y). 52 Chapter 2. Types of Soft Set Remark 2.30.2.If one does not intend to use an algebraic operation on the parameter set, then a double-framed soft set may be taken simply as a pair of mapsα, β:A→P(U)(i.e., the triple〈(α, β);A〉) without imposing the above∗-compatibility axioms. Example 2.30.3(Real-life example of a double-framed soft set: loan pre-screening with positive and negative evidence).LetUbe a set of loan applicants: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . LetAbe a set of screening parameters: A=a Inc , a Cred , a IncCred . Interpreta Inc =“high income”,a Cred =“good credit history”, anda IncCred =“high income and good credit”. Define a binary operation∗:A×A→A(parameter combination) by a Inc ∗a Cred =a IncCred ,a Cred ∗a Inc =a IncCred ,x∗x=x(x∈A). Define two set-valued mapsα, β:A→P(U)as follows. For each parameterx∈A: •α(x)collects applicants withpositive evidencesupportingx, •β(x)collects applicants withnegative evidenceagainstx. Assume the bank’s initial data yield: α(a Inc ) =u 1 , u 2 , u 4 ,β(a Inc ) =u 3 , u 5 , α(a Cred ) =u 1 , u 3 , u 6 ,β(a Cred ) =u 2 , u 4 , and for the combined parameter take α(a IncCred ) =u 1 ,β(a IncCred ) =u 2 , u 3 , u 4 , u 5 . Then the compatibility conditions hold: α(a IncCred )⊇α(a Inc )∩α(a Cred ) =u 1 , u 2 , u 4 ∩u 1 , u 3 , u 6 =u 1 , and β(a IncCred )⊆β(a Inc )∪β(a Cred ) =u 3 , u 5 ∪u 2 , u 4 =u 2 , u 3 , u 4 , u 5 . Hence〈(α, β);A〉is a double-framed soft set overU. Interpretation:αlists candidates supported by a criterion,βlists candidates contradicted by it, and the operation∗combines criteria so that positive support becomes at least as strict (intersection), while negative evidence is at most as broad (union). 53 Chapter 2. Types of Soft Set 2.31 Bijective Soft Set A bijective soft set partitions the universe into disjoint parameter blocks, covering all elements, assigning each uniquely to one parameter [130–132]. As an extension, concepts such as bijective HyperSoft sets [133–135] have also been studied. Definition 2.31.1(Bijective soft set).[130–132] LetUbe a nonempty universe and letEbe a set of parameters. Asoft setoverUis a pair(F, B), whereB⊆Eis a nonempty parameter set andF:B→P(U). The soft set(F, B)is called abijective soft setif the family of subsetsF(e) e∈B forms a partition ofU, i.e., (B1)Covering: ⋃ e∈B F(e) =U; (B2)Pairwise disjointness:for alle 1 , e 2 ∈Bwithe 1 6 =e 2 , one hasF(e 1 )∩F(e 2 ) =∅; (B3)Nontrivial blocks (optional but standard for literal bijectivity):F(e)6 =∅for all e∈B. Equivalently, for everyu∈Uthere exists auniqueparametere∈Bsuch thatu∈F(e). If we denote the image family by Y:=F(e)|e∈B ⊆ P(U), then (under (B3)) the mapF:B→Yis a bijection. Remark 2.31.2.Many papers state bijective soft sets using only (B1)–(B2). In that case, literal bijectivity holds after discarding any parameters with empty images: setB ∗ :=e∈B|F(e)6= ∅and restrictFtoB ∗ . ThenF(e) e∈B ∗ is a partition ofUandF:B ∗ → F(e)|e∈B ∗ is bijective. Example 2.31.3(Real-life example of a bijective soft set: unique department assignment).Let Ube the set of employees in a company: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 . LetEbe a set of parameters and choose a nonempty subset B=e HR , e ENG , e FIN , e MKT ⊆E, where each parameter denotes a department: e HR =Human Resources, e ENG =Engineering, e FIN =Finance, e MKT =Marketing. 54 Chapter 2. Types of Soft Set Define a mappingF:B→P(U)by the employees assigned to each department: F(e HR ) =u 1 , u 6 ,F(e ENG ) =u 2 , u 3 , u 7 , F(e FIN ) =u 4 ,F(e MKT ) =u 5 . Then ⋃ e∈B F(e) =U(every employee belongs to some department), and the setsF(e HR ), F(e ENG ), F(e FIN ), F(e MKT ) are pairwise disjoint (no employee belongs to two departments simultaneously). Moreover, each F(e)is nonempty. HenceF(e) e∈B is a partition ofU, so(F, B)is a bijective soft set. Equivalently, each employee u∈Uhas a unique department-parametere∈Bsuch thatu∈F(e). 2.32 Ranked Soft Set Ranked soft sets map each parameter to an ordered partition of the universe, expressing graded satisfaction levels for uncertain decision-making. The definition of the Ranked Soft Set is de- scribed as follows [136]. Definition 2.32.1(Ranked Soft Set).[136] LetUbe a nonempty finite universe and letEbe a set of parameters. Aranked partitionofUis an ordered collection V= (V 0 , V 1 , . . . , V k ) of subsets ofUsatisfying: 1.V 0 ∪V 1 ∪·∪V k =U, 2. For indicesi, jwith0≤i < j≤k, the elements inV j are regarded as satisfying the corresponding attribute with a higher degree than those inV i . Aranked soft setoverUis a pair(R, E)where R:E→R(U) is a mapping from the set of parametersEto the familyR(U)of all ranked partitions ofU. That is, for eacht∈E,R(t)is a ranked partition ofUrepresenting the graded evaluation of the propertyton the elements ofU. Example 2.32.2(Real-life example of a ranked soft set: hotel recommendation by cleanliness). LetUbe a finite set of hotels in a city: U=h 1 , h 2 , h 3 , h 4 , h 5 . LetEbe a set of evaluation parameters and consider the parameter t=“cleanliness”∈E. 55 Chapter 2. Types of Soft Set Assume we use four ordered satisfaction levels (from worst to best), sok= 3and we form a ranked partition R(t) = (V 0 , V 1 , V 2 , V 3 ), where eachV j ⊆Ucollects hotels assessed at rankj, and higherjmeans cleaner. For instance, based on recent inspection reports and user reviews, suppose we obtain: V 0 =h 4 (poor cleanliness),V 1 =h 2 (fair cleanliness), V 2 =h 3 , h 5 (good cleanliness),V 3 =h 1 (excellent cleanliness). ThenV 0 ∪V 1 ∪V 2 ∪V 3 =U, and for0≤i < j≤3, the hotels inV j are regarded as satisfying the parametertmore strongly than those inV i . Define a ranked soft setR:E→R(U)by specifying such a ranked partition for each parameter. In particular, the valueR(t)above encodes a graded evaluation ofcleanlinessonU, which can be used to recommend hotels by prioritizing higher-ranked blocks (e.g., selecting fromV 3 first, thenV 2 , etc.). 2.33 Refined Soft Set Refined soft sets index multiple soft sets by secondary parameters, representing several eval- uators’ mappings from common attributes to subsets simultaneously. Related notions include refined neutrosophic sets[137–139]. The definition of the Refined Soft Set is described as fol- lows [140–142]. Definition 2.33.1(Refined Soft Set).[140] LetUbe a nonempty universe, and letEandF be two sets of parameters with E∩F=∅. LetAbe a nonempty subset ofE. For each parameterb∈F, let f b :A→P(U) be a soft set overU; that is, for eacha∈A, we havef b (a)⊆U. Then the collection (f b , A)|b∈F is called arefined soft setoverUwith respect to the parameter setAand the indexing setF. Equivalently, a refined soft set may be viewed as a mapping f:F→soft sets overUwith parameter setA, defined byf(b) =f b for allb∈F. Example 2.33.2(Real-life example of a refined soft set: multi-expert product screening).Let Ube a set of job applicants for a software engineer position: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 . 56 Chapter 2. Types of Soft Set LetEbe a parameter set of evaluation criteria and take a nonempty subset A=α 1 , α 2 , α 3 ⊆E, where α 1 =“strong algorithms”,α 2 =“cloud experience”,α 3 =“good communication”. LetFbe an indexing set ofevaluators(secondary parameters), disjoint fromE: F=β 1 , β 2 , β 3 ,E∩F=∅, where β 1 =HR,β 2 =Engineering manager,β 3 =Senior engineer. For each evaluatorβ∈F, define a soft set f β :A−→P(U), wheref β (α)is the subset of applicants judged by evaluatorβto satisfy criterionα. For instance, suppose the assessments are: f β 1 (α 1 ) =u 1 , u 2 , u 4 , f β 1 (α 2 ) =u 2 , u 3 , u 5 , f β 1 (α 3 ) =u 1 , u 3 , u 6 , f β 2 (α 1 ) =u 1 , u 4 , u 5 , f β 2 (α 2 ) =u 2 , u 5 , u 6 , f β 2 (α 3 ) =u 1 , u 2 , u 6 , f β 3 (α 1 ) =u 2 , u 4 , u 6 , f β 3 (α 2 ) =u 1 , u 3 , u 6 , f β 3 (α 3 ) =u 1 , u 4 , u 5 . Then the collection (f β , A)|β∈F is arefined soft setoverU: it records, for the same primary criteria setA, multiple soft sets indexed by evaluators inF. Equivalently, it is the mapping f:F−→soft sets overUwith parameter setA,f(β) =f β . In practice, such a refined soft set supports consensus or aggregation rules (e.g., selecting appli- cants satisfyingα 1 according to at least two evaluators). 2.34 MultiSoft Set A MultiSoft set maps multiple parameters to subsets of the universe, representing simultaneous parameterized approximations for decision-making tasks [143–145]. Related notions also include multi-fuzzy sets[146,147] andmulti-neutrosophic sets[148,149]. Definition 2.34.1(Multisoft set).[143–145] LetUbe a nonempty universe of discourse and letEbe a (nonempty) set of parameters. LetA⊆Ebe a (nonempty) subset of parameters, and writeP(U)for the power set ofU. A pair(F, A)is called amultisoft setoverUif F:A−→P(U). For each parametera∈A, the subsetF(a)⊆Uis thea-approximation(ora-value set) of the multisoft set(F, A). 57 Chapter 2. Types of Soft Set Example 2.34.2(Real-life example of a multisoft set: smartphone selection by multiple crite- ria).LetUbe a finite set of smartphone models: U=s 1 , s 2 , s 3 , s 4 , s 5 , s 6 . LetEbe a set of decision parameters and take the nonempty subset A=a Cam , a Batt , a Price ⊆E, wherea Cam =“good camera”,a Batt =“long battery life”, anda Price =“affordable price”. Define a mappingF:A→ P(U)by listing the models that satisfy each criterion according to reviews and specifications: F(a Cam ) =s 1 , s 3 , s 5 ,F(a Batt ) =s 2 , s 3 , s 6 ,F(a Price ) =s 1 , s 2 , s 4 . Then(F, A)is a multisoft set overU. Interpretation: the familyF(a) a∈A provides multiple parameterized approximations ofU, supporting multi-criteria selection such as choosing phones inF(a Cam )∩F(a Batt )(good camera andlong battery life). 2.35 GraphicSoft Set A GraphicSoft Set further generalizes this framework by mapping each subgraph of an attribute graph to a subset of the universe, thereby embedding inter‐attribute relationships into the soft‐set model [150]. Definition 2.35.1(GraphicSoft Set).[150] LetUbe a universe of discourse, and letG= (V, E) be a graph representing a set of attributes and their relationships. AGraphicSoft Setis defined as a mapping F:P(G)→P(U), which assigns to each subgraphH∈ P(G)a subsetF(H)⊆U. Intuitively,F(H)represents the set of objects inUthat possess the combined attributes described by the subgraphH. Example 2.35.2(Real-life example of a GraphicSoft Set: diet-oriented product search).LetU be a small catalog of packaged foods: U=p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , wherep 1 = tofu bowl,p 2 = chicken salad,p 3 = gluten-free quinoa crackers,p 4 = vegan protein bar,p 5 = Greek yogurt,p 6 = oat milk. LetG= (V, E)be anattribute graphwhose vertices represent dietary attributes: V=v V , v G , v L , v P , wherev V = Vegan,v G = Gluten-free,v L = Low-sugar,v P = High-protein. Assume the edges encodecompatibility/certificationrequirements between attributes: E= v V , v G ,v V , v L ,v L , v P . 58 Chapter 2. Types of Soft Set For each vertexv∈V, define the set of products having attributev: S(v)⊆U, and for each edgev, w ∈E, define the set of products satisfying thejoint requirement(e.g., certified compatible) for(v, w): C(v, w)⊆U. For concreteness, take S(v V ) =p 1 , p 4 , p 6 , S(v G ) =p 1 , p 3 , p 4 , S(v L ) =p 3 , p 4 , p 5 , S(v P ) =p 1 , p 2 , p 4 , p 5 , and C(v V , v G ) =p 1 , p 4 , C(v V , v L ) =p 4 , C(v L , v P ) =p 4 , p 5 . Define a mappingF:P(G)→ P(U)as follows. For any subgraphH= (V(H), E(H))ofG, set F(H) := ( ⋂ v∈V(H) S(v) ) ∩ ( ⋂ v,w∈E(H) C(v, w) ) , with the convention that the intersection over an empty family equalsU. ThenF(H)returns the products satisfying the attributes inV(H)together with the relational constraints inE(H). For instance: 1. IfH 1 hasV(H 1 ) =v V , v G andE(H 1 ) =v V , v G , then F(H 1 ) =S(v V )∩S(v G )∩C(v V , v G ) =p 1 , p 4 . 2. IfH 2 hasV(H 2 ) =v V , v L , v P andE(H 2 ) =v V , v L ,v L , v P , then F(H 2 ) =S(v V )∩S(v L )∩S(v P )∩C(v V , v L )∩C(v L , v P ) =p 4 . Thus(F, G)is a GraphicSoft Set modeling diet-oriented retrieval where edges encode compati- bility/certification relations among attributes. For reference, a comparison between Soft Sets and GraphicSoft Sets is provided in Table 2.6. A related concept is that one can define aDAG-soft setas follows. A directed acyclic graph (DAG) is a directed graph containing no directed cycles, enabling topological ordering of vertices for computation [151–154]. Definition 2.35.3(DAGSoft set (acyclic-hierarchical parameter soft set)).LetUbe a nonempty universe and letD= (A,→)be a finite directed acyclic graph (DAG) of parameters. ADAGSoft setoverUis a soft set(F, A),F:A→P(U), satisfying the hierarchical coherence rule a→b=⇒F(b)⊆F(a) (a, b∈A). 59 Chapter 2. Types of Soft Set Table 2.6: Concise comparison between Soft Sets and GraphicSoft Sets over a universeU. AspectSoft SetGraphicSoft Set UniverseFixed universe of discourseU.Same: fixed universeU. Parameter carrierA (usually finite) parameter setS⊆A (attributes). An attributegraphG= (V, E)encoding attributes (V) and their relations (E). Indexing domain of the map Parameterse∈S.SubgraphsH∈Sub(G)(or equivalently P(G)). Basic data (defini- tion) A mapF:S→ P(U); the soft set is (F, S). A mapF:Sub(G)→ P(U); the struc- ture is(F, G). What the “param- eter” means A single attribute/criterione(treated independently unless combined exter- nally). Apattern of interacting attributes: a subgraphH(vertices + edges) rep- resents a chosen combination together with their relationships. Howattribute- relationsare represented Not represented intrinsically (no edges/constraints among parameters in the basic model). Represented intrinsically viaE(H): edges record dependency, compatibility, adjacency, etc. Admissible combi- nations Typically handled by taking finite fam- ilies of parameters (e.g., decision rules using multiplee’s), but this isexternal toF. Built-in: any subgraphHis an admis- sible “combined parameter”;F(H)di- rectly models objects satisfying the com- bined/related attributes encoded byH. Granularity / ex- pressiveness “Unary” parameterization:Fassigns sets to individual parameters. “Structured” parameterization:Fas- signs sets to structured parameter- objects (subgraphs), capturing multi- attribute interactions explicitly. Model size (typi- cal) Depends on|S|evaluations ofF(e).Potentially much larger: requires values for many subgraphs (in worst case expo- nential in|V|+|E|). Natural reduction / embedding Base model.Contains soft-set-like information by re- stricting to subgraphs representing sin- gle attributes (e.g., isolated-vertex sub- graphs). Conversely, a soft set can be viewed as a degenerate case where only “atomic” subgraphs are used. Typical use-casesDecision making with attribute-wise ap- proximations; uncertainty via parame- terized subsets. Decision/knowledge modeling wherere- lations among attributes matter(de- pendencies, synergies, conflicts), and where one wants selections indexed by attribute-interaction patterns. Example 2.35.4(Real-life example of a DAGSoft set: IT helpdesk ticket taxonomy with hier- archical parameters).LetUbe a finite set of IT helpdesk tickets: U=t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 . Consider a parameter DAGD= (A,→)describing an issue taxonomy: A=a Acc , a Conn , a WiFi , a VPN , a Auth , a SSO , where the intended meanings are a Acc =“access problem”, a Conn =“connectivity problem”, a WiFi =“Wi-Fi problem”, a VPN =“VPN problem”, a Auth =“authentication problem”, a SSO =“SSO problem”. Let the directed edges encode refinement (child is more specific): a Acc →a Conn ,a Acc →a Auth ,a Conn →a WiFi ,a Conn →a VPN ,a Auth →a SSO . This digraph is acyclic (a DAG). 60 Chapter 2. Types of Soft Set Define a soft mappingF:A→P(U)by F(a Acc ) =t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 , F(a Conn ) =t 1 , t 2 , t 4 , t 6 , F(a WiFi ) =t 1 , t 2 , F(a VPN ) =t 4 , t 6 , F(a Auth ) =t 3 , t 5 , t 7 , F(a SSO ) =t 5 , t 7 . Then the hierarchical coherence rule in Definition 2.35.3 holds, because each directed edgea→b satisfiesF(b)⊆F(a); for instance, F(a WiFi )⊆F(a Conn )⊆F(a Acc ),F(a SSO )⊆F(a Auth )⊆F(a Acc ). Hence(F, A)is a DAGSoft set overU, representing a parameterized classification of tickets in which more specific issue-types select subsets of the tickets selected by their broader parent types. 2.36 CycleSoft Set A CycleSoft Set extends Soft Sets by organizing parameters in a cycle graph, mapping cycle subgraphs to subsets of a universal set for structured decision-making [155]. Definition 2.36.1(CycleSoft Set).[155] LetUbe a universal set and letC= (A, E C )be a cycle graph, whereAis a set of parameters arranged in a cycle and E C =(a i , a i+1 )|a i , a i+1 ∈A∪(a n , a 1 ) describes the cyclic adjacency among the parameters. Define the power set ofCas P(C) =H|His a subgraph ofC. ACycleSoft Setis a mapping F:P(C)→P(U), where for each subgraphH∈P(C),F(H)⊆Urepresents the set of objects associated with the combination of parameters corresponding toH. A common aggregation is to define, for eachH, F(H)(x) = ⋂ a∈V(H) f(a)(x), withf(a) :U→[0,1](or characteristic functions in the crisp case). Example 2.36.2(Real-life example of a CycleSoft Set: selecting restaurants by cyclic service attributes).LetUbe a set of restaurants: U=r 1 , r 2 , r 3 , r 4 , r 5 , r 6 . Consider four service-related parameters arranged in a cycle: A=a 1 , a 2 , a 3 , a 4 , 61 Chapter 2. Types of Soft Set where a 1 =“good food”, a 2 =“good service”, a 3 =“clean”, a 4 =“good value”. LetC= (A, E C )be the cycle graph with edges E C =(a 1 , a 2 ),(a 2 , a 3 ),(a 3 , a 4 ),(a 4 , a 1 ). LetP(C)denote the family of all subgraphs ofC. First define a (crisp) soft mappingf:A→P(U)by f(a 1 ) =r 1 , r 2 , r 5 , f(a 2 ) =r 1 , r 3 , r 5 , r 6 , f(a 3 ) =r 2 , r 3 , r 4 , r 5 , f(a 4 ) =r 1 , r 2 , r 4 . For a subgraphH∈P(C), define the CycleSoft mappingF:P(C)→P(U)by F(H) := ⋂ a∈V(H) f(a), with the conventionF(∅) =U. For example, letH 1 be the path subgraph with verticesa 1 , a 2 , a 3 and edges(a 1 , a 2 ),(a 2 , a 3 ). Then F(H 1 ) =f(a 1 )∩f(a 2 )∩f(a 3 ) =r 1 , r 2 , r 5 ∩r 1 , r 3 , r 5 , r 6 ∩r 2 , r 3 , r 4 , r 5 =r 5 . Thusr 5 is the (unique) restaurant satisfying the combined cyclic attribute pattern “good food →good service→clean”. HenceFis a CycleSoft Set onUindexed by subgraphs of the cycleC. 2.37 ClusterSoft Set A ClusterSoft Set groups multiple Soft Sets, capturing relationships among clustered attributes and mapping them to subsets of a universal set for decision modeling [155]. Definition 2.37.1(ClusterSoft Set).[155] LetF i i∈I be a finite family of soft sets over a universeU, where each soft setF i is a mapping F i :A i →P(U) for some set of attributesA i . Suppose the index setIis partitioned into clustersC j j∈J with eachC j ⊆IandC j ∩C k =∅forj6 =k. AClusterSoft Setis defined as a mapping G:C j :j∈J→P(U) given by G(C j ) = ⋃ i∈C j F ∗ i (A i ), whereF ∗ i (A i )denotes the set of objects inUassociated with soft setF i (possibly after appro- priate aggregation or normalization). The union is taken in the usual set-theoretic sense. 62 Chapter 2. Types of Soft Set Example 2.37.2(Real-life example of a ClusterSoft Set: grouping customer segments from multiple marketing soft sets).LetUbe a set of customers of an online shop: U=u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 . Assume the marketing team maintains a finite family of soft setsF i i∈I , each capturing cus- tomers associated with certain attributes in a specific campaign. Let I=1,2,3,4. Define, for eachi∈I, a parameter setA i and a soft mappingF i :A i →P(U). Campaign 1 (sports campaign).LetA 1 =a Run , a Gym and suppose F 1 (a Run ) =u 1 , u 3 , u 6 ,F 1 (a Gym ) =u 2 , u 3 , u 7 . Campaign 2 (outdoor campaign).LetA 2 =a Hike and suppose F 2 (a Hike ) =u 1 , u 4 , u 6 , u 8 . Campaign 3 (family campaign).LetA 3 =a Kids and suppose F 3 (a Kids ) =u 2 , u 5 , u 7 . Campaign 4 (premium campaign).LetA 4 =a Premium and suppose F 4 (a Premium ) =u 3 , u 4 , u 8 . For each soft setF i , define the associated customer set F ∗ i (A i ) := ⋃ a∈A i F i (a)⊆U. Then F ∗ 1 (A 1 ) =u 1 , u 2 , u 3 , u 6 , u 7 , F ∗ 2 (A 2 ) =u 1 , u 4 , u 6 , u 8 , F ∗ 3 (A 3 ) =u 2 , u 5 , u 7 , F ∗ 4 (A 4 ) =u 3 , u 4 , u 8 . Now partition the index setIinto clusters (segments): C Active =1,2,C Lifestyle =3,4,C Active , C Lifestyle is a partition ofI. Define the ClusterSoft mappingG:C Active , C Lifestyle →P(U)by G(C) = ⋃ i∈C F ∗ i (A i ). Hence G(C Active ) =F ∗ 1 (A 1 )∪F ∗ 2 (A 2 ) =u 1 , u 2 , u 3 , u 4 , u 6 , u 7 , u 8 , G(C Lifestyle ) =F ∗ 3 (A 3 )∪F ∗ 4 (A 4 ) =u 2 , u 3 , u 4 , u 5 , u 7 , u 8 . ThusGis a ClusterSoft Set: each cluster aggregates the customers associated with the soft sets belonging to that cluster, producing unified target segments for marketing actions. 63 Chapter 2. Types of Soft Set 2.38 Soft Expert Set Asoft expert setincorporates expert participation and their opinions: parameter–expert–opinion triples are mapped to subsets ofU, equivalently yielding a mapping fromE×X×OintoP(U) [156–159]. As extensions,fuzzy soft expert sets[160,161],neutrosophic soft expert sets[162–164], andHyperSoft expert sets[25,135,165] are also known. Definition 2.38.1(Soft Expert Set).[156–159] LetUbe a universe,Ea set of parameters, Xa set of experts, andO=0,1a set of opinions. PutZ:=E×X×Oand letA⊆Zbe nonempty. Asoft expert setonUis a pair(G, A)where G:A−→P(U) assigns to each tripleα= (e, x, o)∈Aa subsetG(α)⊆U. ThusG(α)collects the elements of Usupported by expertx, with opiniono, under parametere. Example 2.38.2(Real-life example of a soft expert set: smartphone selection by multiple experts).LetUbe a set of smartphone models: U=s 1 , s 2 , s 3 , s 4 , s 5 . LetEbe a set of evaluation parameters: E=e Cam , e Batt , e Price , wheree Cam =“good camera”,e Batt =“long battery life”, ande Price =“affordable price”. LetXbe a set of experts: X=x 1 , x 2 , x 3 , wherex 1 is a reviewer,x 2 is an engineer, andx 3 is a budget-conscious user. LetO=0,1be the set of opinions, where1meansapproveand0meansdisapprove. SetZ=E×X×Oand chooseA=Z(i.e., all expert–parameter–opinion triples are allowed). Define a mappingG:A→P(U)by listing, for each triple(e, x, o), the phones receiving opinion ofrom expertxunder parametere. For example, suppose: G(e Cam , x 1 ,1) =s 1 , s 3 ,G(e Cam , x 1 ,0) =s 2 , s 4 , s 5 , G(e Batt , x 2 ,1) =s 2 , s 3 , s 5 ,G(e Batt , x 2 ,0) =s 1 , s 4 , G(e Price , x 3 ,1) =s 2 , s 4 ,G(e Price , x 3 ,0) =s 1 , s 3 , s 5 , and defineG(e, x, o)similarly for all remaining triples inA. Then(G, A)is a soft expert set onU: each triple(e, x, o)determines the subset of smartphones supported (ifo= 1) or rejected (ifo= 0) by expertxwith respect to parametere. This structure supports aggregation rules such as selecting phones approved by a majority of experts under key parameters. 64 Chapter 2. Types of Soft Set 2.39 Soft Rough Set A soft rough set approximates a subset using a soft set: lower includes certainly contained elements, upper includes possibly related elements [16,166]. Related notions such asfuzzy soft rough sets[167–169] andneutrosophic soft rough sets[166,170,171] are also known. Definition 2.39.1(Soft Rough Set).[16,166] LetS= (F, A)be a soft set over a universeU, and letP= (U, S)be the correspondingsoft approximation space. ForX⊆U, thesoftP-lower andsoftP-upperapproximations ofXare defined by apr P (X) =u∈U:∃a∈Asuch thatu∈F(a)⊆X, apr P (X) =u∈U:∃a∈Asuch thatu∈F(a)andF(a)∩X6=∅. The pair(apr P (X),apr P (X))is called thesoft rough setofXwith respect toP. Example 2.39.2(Real-life example of a soft rough set: identifying “reliable suppliers” from parameterized checklists).LetUbe a set of candidate suppliers: U=s 1 , s 2 , s 3 , s 4 , s 5 . LetAbe a set of audit parameters (checklists): A=a ISO , a OnTime , a LowDefect . Define a soft setS= (F, A)overUby F(a ISO ) =s 1 , s 2 , s 4 (ISO-certified suppliers), F(a OnTime ) =s 1 , s 3 , s 4 (historically on-time suppliers), F(a LowDefect ) =s 2 , s 4 , s 5 (low defect-rate suppliers). LetP= (U, S)be the induced soft approximation space. Suppose the procurement team proposes a target set X=s 1 , s 4 ⊆U ofpreferred suppliers(e.g., shortlisted by a manager). Lower approximation.An elementu∈Ubelongs to apr P (X)if there exists a parameter a∈Asuch thatu∈F(a)⊆X. Here, F(a ISO ) =s 1 , s 2 , s 4 *X, F(a OnTime ) =s 1 , s 3 , s 4 *X, F(a LowDefect ) =s 2 , s 4 , s 5 *X. Thus noF(a)is contained inX, and hence apr P (X) =∅. Upper approximation.An elementu∈Ubelongs to apr P (X)if there existsa∈Asuch that u∈F(a)andF(a)∩X6=∅. Since F(a ISO )∩X=s 1 , s 4 6=∅, F(a OnTime )∩X=s 1 , s 4 6=∅, F(a LowDefect )∩X=s 4 6=∅, every supplier appearing in at least one of theseF(a)sets is possibly compatible withX. Therefore, apr P (X) =F(a ISO )∪F(a OnTime )∪F(a LowDefect ) =s 1 , s 2 , s 3 , s 4 , s 5 =U. Hence the soft rough set ofXwith respect toPis ( apr P (X), apr P (X) ) = (∅, U). Interpretation: based on these coarse checklists, no supplier iscertainlyin the preferred set, while every supplier ispossiblyrelated to it through at least one parameter block. 65 Chapter 2. Types of Soft Set 2.40 Weighted Soft Set A weighted soft set maps each parameter to a universe subset and assigns a weight reflecting that parameter’s relative significance [172–174]. Definition 2.40.1(Weighted Soft Set).[174] LetUbe a finite universe of discourse and letE be a finite set of parameters (attributes). A finiteweighted soft setoverUis defined as a triple (μ, A, Q), where: •A⊆Eis a finite set of selected parameters, •μ:A→P(U)is a mapping that assigns to each parametera∈Aa subsetμ(a)⊆U, and •Q:A→[0,1]is a finite weight function that assigns to each parametera∈Aa weight Q(a), representing its relative importance. Thus, the finite weighted soft set can be represented as (μ, A, Q) =(a, μ(a), Q(a))|a∈A. Example 2.40.2(Real-life example of a weighted soft set: selecting apartments with weighted criteria).LetUbe a finite set of apartments: U=h 1 , h 2 , h 3 , h 4 , h 5 . LetEbe a finite set of decision parameters and choose A=a Rent , a Comm , a Safe ⊆E, wherea Rent =“affordable rent”,a Comm =“short commute”, anda Safe =“safe neighborhood”. Define the soft mappingμ:A→P(U)by listing apartments that satisfy each criterion: μ(a Rent ) =h 1 , h 3 , h 5 ,μ(a Comm ) =h 2 , h 3 , h 4 ,μ(a Safe ) =h 1 , h 2 , h 4 . Assign a weight functionQ:A→[0,1]reflecting the user’s priorities: Q(a Rent ) = 0.50,Q(a Comm ) = 0.30,Q(a Safe ) = 0.20. Then(μ, A, Q)is a weighted soft set overU. Interpretation: affordability is the most important criterion (weight0.50), followed by commute time and safety. A simple weighted choice score for an apartmenth∈Ucan be formed by summing the weights of parameters that accepth, e.g., Score(h) := ∑ a∈A h∈μ(a) Q(a), so that higher scores indicate better overall fit to the weighted criteria. 66 Chapter 2. Types of Soft Set 2.41 Other Soft Set In this section, we briefly describe several other variants of soft sets. • Geometric soft sets [175]: Geometric soft sets map parameters to point-incidence subsets, capturing hyperplane or distance relations, enabling geometric realization and network analysis efficiently. • Multiparameterized soft sets [176, 177]: A multiparameterized soft set maps each multi- parameter tuple to a subset of the universe, capturing combined attribute approximations effectively simultaneously. • Concave Soft sets [178–180]: Concave soft sets are order-preserving: ifx≤ythenF(x)⊆ F(y), equivalentlyFcoincides with its closure[[F]]for the given order. 67 Chapter 3 Uncertain Soft Theory In this chapter, we examine uncertainty-aware soft set models, including fuzzy soft sets and neutrosophic soft sets. 3.1 Fuzzy Soft Set A fuzzy soft set assigns each parameter a fuzzy subset of the universe, giving graded membership degrees for objects under parameters [181–183]. Related notions includefuzzy HyperSoft sets [125,184,185] andneutrosophic HyperSoft sets[186–189]. Definition 3.1.1(Fuzzy Soft Set).[181–183] LetUbe a nonempty universe andEa set of parameters. WriteF(U) :=μ:U→[0,1]for the family of all fuzzy subsets ofU. For a fixed subsetA⊆E, afuzzy soft setoverU(with respect toA) is a pair (Γ A , A),Γ A :A−→F(U), x7−→μ x (·), so that it can be represented as the collection Γ A =(x, μ x )|x∈A, μ x :U→[0,1]. Foru∈Uandx∈A, the valueμ x (u)is the degree to whichuapproximately satisfies the parameterx. Equivalently, one may specify a map ̃ Γ :E→ F(U)with ̃ Γ(x) =0(the zero membership function) for allx/∈A. 3.2 Intuitionistic Fuzzy Soft Set An intuitionistic fuzzy soft set assigns each parameter an intuitionistic fuzzy set, specifying membership and non-membership degrees with hesitation [190–193]. 69 Chapter 3. Uncertain Soft Theory Definition 3.2.1(Intuitionistic fuzzy set (Atanassov)).[5,194] LetUbe a nonempty universe. Anintuitionistic fuzzy set(IFS)AonUis specified by a pair of functions μ A :U→[0,1],ν A :U→[0,1], called themembershipandnon-membershipfunctions, respectively, such that 0≤μ A (u) +ν A (u)≤1for allu∈U. Thehesitation (indeterminacy) degreeis then π A (u) := 1−μ A (u)−ν A (u)∈[0,1]. We writeIFS(U)for the class of all intuitionistic fuzzy sets onU. Definition 3.2.2(Intuitionistic fuzzy soft set).[190,191] LetUbe a nonempty universe and letEbe a nonempty set of parameters. Anintuitionistic fuzzy soft set(IFSS) overU(with parameter setE) is a pair ( ̃ F , A),A⊆E, where ̃ F:A−→IFS(U) is a mapping. Equivalently, an IFSS can be represented as a family of ordered pairs ( ̃ F , A) = (e, ̃ F(e)) ∣ ∣ e∈A, ̃ F(e)∈IFS(U) . If one prefers to keep the full parameter setEas the domain, one may extend ̃ Fto a mapping ̃ F:E→IFS(U)by setting, for eache∈E , thenull IFS ̃ F(e) = ̃ ∅defined by μ ̃ ∅ (u) = 0,ν ̃ ∅ (u) = 1 (∀u∈U), which matches the common convention in the literature. 3.3 Neutrosophic Soft Set A neutrosophic set assigns each element independent truth, indeterminacy, and falsity degrees, generalizing fuzzy and intuitionistic fuzzy sets [8, 9]. The Neutrosophic Soft Set is a concept that combines the principles of Neutrosophic Sets and Soft Sets [137,195–201]. The definition is provided below. Definition 3.3.1(Neutrosophic Soft Set [202, 203]).LetUbe a universe andEa set of pa- rameters. ANeutrosophic Soft Set (NSS)overUis defined as a pair ( F, A ) , whereA⊆E and F:A−→P(U), withP(U)being the collection ofNeutrosophic SetsonU. Hence for each parametere∈A, F(e) = ( T F(e) , I F(e) , F F(e) ) is a Neutrosophic Set onU, satisfying 0≤T F(e) (x) +I F(e) (x) +F F(e) (x)≤3,∀x∈U. 70 Chapter 3. Uncertain Soft Theory 3.4 Plithogenic Soft Set A plithogenic set models multi-attribute membership using contradiction degrees among at- tribute values, generalizing fuzzy, intuitionistic, and neutrosophic sets [14, 204]. A plithogenic soft set maps attribute-value tuples to plithogenic evaluations and corresponding subsets, incor- porating contradiction profiles relative to dominant values [205,206]. Definition 3.4.1(Plithogenic soft set).[205, 206] LetUbe a universe of discourse and let z∈C, F, IF, N. We writeP z (U)for thez-power setofU, defined by P C (U) :=P(U),P F (U) :=μ:U→[0,1], P IF (U) := (μ, ν) :U→[0,1] 2 ∣ ∣ 0≤μ(u) +ν(u)≤1 (∀u∈U) , and letP N (U)denote the family of (single-valued) neutrosophic sets onU(i.e., triples(T, I, F) : U→[0,1] 3 satisfying the usual neutrosophic constraints, according to the chosen convention). Leta 1 , a 2 , . . . , a n (n≥1) be distinct attributes with corresponding (pairwise disjoint) value sets V 1 , V 2 , . . . , V n such thatV i ∩V j =∅fori6 =j. Set Υ :=V 1 ×V 2 ×·×V n . Fix adominant value vectorD= (D 1 , . . . , D n )∈Υand, for eachi∈1, . . . , n, acontradiction degree function c i :V i ×V i −→[0,1]satisfyingc i (v, v) = 0, c i (v, w) =c i (w, v). Define [0,1] D := [0,1] n and, forυ= (v 1 , . . . , v n )∈Υ, define the contradiction vector relative toDby c D (υ) := ( c 1 (D 1 , v 1 ), c 2 (D 2 , v 2 ), . . . , c n (D n , v n ) ) ∈[0,1] D . Az-plithogenic soft set(briefly,plithogenic soft set) overUis a pair(F z P ,Υ)where F z P : Υ−→[0,1] D ×P z (U). Equivalently, for eachυ∈Υwe can write F z P (υ) = ( c D (υ), S υ ) ,withS υ ∈P z (U), so that each attribute-value tupleυis assigned (i) its contradiction profile relative to the domi- nant tupleD, and (i) az-valued subsetS υ of the universeU. 71 Chapter 3. Uncertain Soft Theory 3.5 Uncertain Soft Set An Uncertain Set is any set-theoretic model assigning graded, possibly multi-component member- ship degrees to elements, generalizing fuzzy, intuitionistic, neutrosophic, plithogenic and related uncertainty frameworks unified [207–209]. Definition 3.5.1(Uncertain Set).[207] LetUbe the collection of all Uncertain Models. Fix U∈U,Dom(U)⊆[0,1] r for some integerr≥1, and letXbe a nonempty base set (universe of discourse). AnUncertain Set of typeUonXis a pair A U = (X, μ), where μ:X−→Dom(U) assigns to each elementx∈XaU–membership degree μ(x)∈Dom(U). Equivalently, once the base setXand the Uncertain ModelUare fixed, we may identify the Uncertain Set with its membership function and simply write A U :X−→Dom(U), x7−→μ(x), and view the collection of all Uncertain Sets of typeUonXas the function space ( Dom(U) ) X =μ|μ:X→Dom(U). In this sense, an Uncertain Set is aU–labeling of the base setXby membership–degree tuples taken from Dom(U). Remark 3.5.2(Recovery of classical fuzzy–type sets).LetXbe a nonempty set and letA U = (X, μ)be an Uncertain Set of typeU. 1. (Fuzzy Set) TakeU=Fuzzy with Dom(U) = [0,1] = [0,1] 1 . Then an Uncertain Set of typeUis exactly a fuzzy set in the sense of Zadeh, since μ:X→[0,1] is the usual fuzzy membership function. 72 Chapter 3. Uncertain Soft Theory 2. (Intuitionistic Fuzzy Set) TakeU=Intuitionistic Fuzzy with Dom(U) = (μ, ν)∈[0,1] 2 |μ+ν≤1 ⊆[0,1] 2 . ThenA U = (X, μ)coincides with an intuitionistic fuzzy set, because for eachx∈X, μ(x) = ( μ A (x), ν A (x) ) ∈[0,1] 2 satisfiesμ A (x) +ν A (x)≤1. 3. (Neutrosophic Set) TakeU=Neutrosophic with Dom(U) = (T, I, F)∈[0,1] 3 |0≤T+I+F≤3 ⊆[0,1] 3 . ThenA U = (X, μ)is exactly a single–valued neutrosophic set, since μ(x) = ( T A (x), I A (x), F A (x) ) ∈[0,1] 3 with0≤T A (x) +I A (x) +F A (x)≤3for allx∈X. 4. (Plithogenic Set) For a Plithogenic ModelU=Plithogenic with degree–domain Dom(U) = ( v,pdf(x, v),pCF(v 1 , v 2 ) ) ∣ ∣ ∣ v∈P v ,pdf(x, v)∈[0,1] s ,pCF(v 1 , v 2 )∈[0,1] t ⊆[0,1] s+t+` , an Uncertain Set of typeUonXreproduces a Plithogenic Set onX, since each μ(x)∈Dom(U) encodes the Plithogenic degrees associated withx∈X. Thus, by choosing different Uncertain ModelsU∈Uand their corresponding domains Dom(U)⊆ [0,1] r , the general notion of an Uncertain Set in Definition 3.5.1 unifies fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, plithogenic sets, and many other existing uncertainty–set frame- works. Definition 3.5.3(Uncertainty domain andD-uncertain sets).LetUbe a nonempty universe. Fix an integerr≥1and a nonempty set (degree-domain) D⊆[0,1] r . AD-uncertain setonUis a mapping μ:U−→D. We denote the class of allD-uncertain sets onUby Unc D (U) :=D U =μ|μ:U→D. Definition 3.5.4(Uncertain soft set).LetUbe a nonempty universe and letEbe a nonempty set of parameters. Fix a nonempty uncertainty domainD⊆[0,1] r as in Definition 3.5.3. Let A⊆Ebe nonempty. AD-uncertain soft set(briefly, anuncertain soft set) overUwith parameter setAis a pair (F, A)where F:A−→Unc D (U). Equivalently, for each parametere∈A, the valueF(e)is aD-uncertain set onU, i.e., a function F(e) :U−→D,u7−→F(e)(u)∈D. 73 Chapter 3. Uncertain Soft Theory Remark 3.5.5(Two standard choices ofD).(i)Fuzzy domain:D Fuz := [0,1]⊆[0,1] 1 . (i)Single-valued neutrosophic domain: D Neu := (t, i, f)∈[0,1] 3 ∣ ∣ 0≤t+i+f≤3 ⊆[0,1] 3 . (If one adopts a different conventional constraint for neutrosophic triples, replaceD Neu ac- cordingly; the theorem below remains valid.) Theorem 3.5.6(Uncertain soft sets generalize fuzzy and neutrosophic soft sets).LetUbe a universe,Ea parameter set, andA⊆Enonempty. (1)(Fuzzy soft sets as a special case) IfD=D Fuz = [0,1], thenD-uncertain soft sets(F, A) are exactly fuzzy soft sets overUwith parameter setA. (2)(Neutrosophic soft sets as a special case) IfD=D Neu , thenD-uncertain soft sets(F, A) are exactly (single-valued) neutrosophic soft sets overUwith parameter setA. Proof.(1)AssumeD= [0,1]. Then Unc D (U) =D U = [0,1] U , the set of all membership functions onU. Hence an uncertain soft set is a map F:A→[0,1] U . For eache∈A, putμ e :=F(e)∈[0,1] U . Thus(F, A)assigns to each parameterea fuzzy subsetμ e ofU, which is precisely the definition of a fuzzy soft set. Conversely, any fuzzy soft set(Γ, A)withΓ(e) =μ e :U→[0,1]is exactly a mapΓ :A→[0,1] U =Unc D (U), hence a D-uncertain soft set. Therefore the two notions coincide whenD= [0,1]. (2)AssumeD=D Neu ⊆[0,1] 3 . ThenUnc D (U) =D U is the class of all functions U→D Neu , u7→(T(u), I(u), F(u)), i.e., the class of (single-valued) neutrosophic sets onUunder the chosen constraint. An uncertain soft set is a map F:A→D U Neu , so eache∈Ais assigned a neutrosophic setF(e)onU. This is exactly the definition of a (single- valued) neutrosophic soft set. Conversely, any neutrosophic soft set(G, A)is by definition a map G:A→D U Neu =Unc D (U), hence aD-uncertain soft set. Thus the two notions coincide when D=D Neu . Remark 3.5.7(Optional extension: parameter-dependent uncertainty types).If one wishes to allow different uncertainty domains per parameter, one may fix a familyD e ⊆[0,1] r e e∈A and define an “untyped” uncertain soft set as a mapF(e) :U→D e . The specialization arguments in Theorem 3.5.6 then apply componentwise. 74 Chapter 3. Uncertain Soft Theory 3.6 Z-Soft Set A Z-soft set maps each parameter and object to a Z-number, combining fuzzy assessment with fuzzy reliability information. Definition 3.6.1(Fuzzy number).[210,211] Afuzzy numberon[0,1]is a fuzzy set ̃ A: [0,1]→ [0,1]such that: 1. ̃ Aisnormal: sup x∈[0,1] ̃ A(x) = 1; 2. ̃ Aisconvex: for everyα∈(0,1], theα-cut [ ̃ A] α :=x∈[0,1]| ̃ A(x)≥α is a (possibly degenerate) closed interval; 3. ̃ Ais upper semicontinuous and has compact support in[0,1]. We writeFN([0,1])for the family of all fuzzy numbers on[0,1]. Definition 3.6.2(Z-number).[17] AZ-numberis an ordered pair Z= ( ̃ A, ̃ B), where ̃ A∈FN([0,1])represents a fuzzy restriction (e.g., a fuzzy assessment of a degree), and ̃ B∈FN([0,1])represents the reliability (certainty) of ̃ A. Let Z:=FN([0,1])×FN([0,1]) denote the set of all such Z-numbers. Definition 3.6.3(Z-valued fuzzy set).LetUbe a nonempty universe. AZ-valued fuzzy seton Uis a mapping μ:U−→Z. We denote the family of all Z-valued fuzzy sets onUby ZFS(U) :=Z U . Definition 3.6.4(Z-soft set).LetUbe a nonempty universe and letEbe a nonempty set of parameters. LetA⊆Ebe nonempty. AZ-soft setoverUwith parameter setAis a pair(F, A) where F:A−→ZFS(U). Equivalently,(F, A)may be identified with a single mapping μ:A×U−→Z,μ(e, u) = ( ̃ A e,u , ̃ B e,u ) , where ̃ A e,u encodes the (fuzzy) assessment ofuunder parametere, and ̃ B e,u encodes the relia- bility of that assessment. 75 Chapter 3. Uncertain Soft Theory 3.7 Functorial Soft Set Functorial Sets provide an additional categorical layer: they organize such assignments across objects and morphisms so that structure transport is systematic and compositional [207, 212]. A functorial soft set treats parameters as objects in a category, assigning subsets functorially so morphisms induce consistent transformations naturally. Definition 3.7.1(Functorial Set).[207] LetCbe a category and let F:C −→Set be a covariant functor. The pair(C, F)is called aFunctorial Set. For each objectX∈Ob(C), the setF(X)is interpreted as the collection ofF-structures attached toX. Every morphism f:X→Yinduces a structure-preserving map F(f):F(X)−→F(Y), such thatF(id X ) =id F(X) and F(g◦f) =F(g)◦F(f) for all composable morphismsf, ginC. Definition 3.7.2(Covariant power-set functor).LetSetbe the category of sets and functions. Thecovariant power-set functor P:Set−→Set is defined on objects byP(X) =A|A⊆Xand on morphismsf:X→Yby thedirect imagemap P(f) :P(X)−→P(Y),P(f)(A) :=f[A] =f(x)|x∈A. ThenP(id X ) =id P(X) andP(g◦f) =P(g)◦P(f). Definition 3.7.3(Functorial soft set).LetCbe a category. Let U:C −→SetandE:C −→Set be covariant functors, interpreted as auniverse functorand aparameter functor, respectively. Afunctorial soft seton(C,U,E)is a natural transformation F:E=⇒P ◦U. Equivalently, it is the data of maps F X :E(X)−→P ( U(X) ) (X∈Ob(C)) such that for every morphismf:X→YinCthe following naturality condition holds: P ( U(f) ) ◦F X =F Y ◦E(f). In elementwise form, for alle∈E(X), U(f) [ F X (e) ] =F Y ( E(f)(e) ) ⊆ U(Y). 76 Chapter 3. Uncertain Soft Theory Theorem 3.7.4(Functorial soft sets generalize soft sets and functorial sets).(1)(Soft sets as a special case) LetCbe the terminal category with one object∗and onlyid ∗ . LetU,E: C →Setbe constant functors withU(∗) =UandE(∗) =E. Then every functorial soft set F:E ⇒P ◦Uis uniquely the same as a classical soft set mapping F:E−→P(U). (2)(Functorial sets embed into functorial soft sets) Let(C, F)be a functorial set, i.e., F:C →Set. DefineU:=FandE:=F, and define components F X :F(X)−→P ( F(X) ) ,F X (x) :=x. ThenF:E ⇒P◦Uis a natural transformation, hence a functorial soft set. Moreover, from this functorial soft set one recovers the original functorial set by taking eitherUorE(both equalF). Proof.(1)In the terminal categoryC, a functorU:C →Setis determined by a single set U:=U(∗), and likewiseEis determined by a single setE:=E(∗). A natural transformation F:E ⇒P ◦U is determined by its single component F ∗ :E−→P(U). Since the only morphism is id ∗ , the naturality condition is automatic. Thus, settingF:=F ∗ yields precisely a soft set mappingF:E→ P(U), and conversely any such mapping defines a unique natural transformation. Hence the notions coincide. (2)Letf:X→Ybe a morphism inCand letx∈F(X). Compute the left-hand side of naturality: P ( U(f) )( F X (x) ) =P ( F(f) ) (x) =F(f)[x] =F(f)(x). Compute the right-hand side: F Y ( E(f)(x) ) =F Y ( F(f)(x) ) =F(f)(x). ThusP(U(f))◦F X =F Y ◦E(f)holds for allf, soFis natural and hence defines a functorial soft set. Finally, sinceU=E=Fby construction, the original functorial set is recovered immediately from the functorial soft set. 77 Chapter 4 Applications of Soft Set In this chapter, we describe several studies on extended concepts developed using soft set theory. 4.1 Soft Graph A soft graph assigns each parameter a subgraph of a fixed graph, modeling uncertain, parameter- ized relationships among vertices and edges [213,214]. As extensions, concepts such as HyperSoft Graphs [215–217], Fuzzy Soft Graphs [218, 219], Neutrosophic Soft Graphs [220, 221], Soft Hy- perGraphs [222, 223], Soft SuperHyperGraphs [224], and Soft Directed Graphs [225, 226] have also been studied. Definition 4.1.1(Soft Graph).[213,214] LetG= (V, E)be a simple undirected graph andC a nonempty set of parameters. Asoft graphoverGwith parameter setCis a quadruple ( G, C, A, B ) , where A:C−→P(V),B:C−→P(E), and for eachc∈C, B(c)⊆ u, v∈E:u∈A(c), v∈A(c) . The pair ( A(c), B(c) ) is called thesoft subgraphat parameterc. Example 4.1.2(Example of a soft graph: a friendship network under different interaction contexts).LetG= (V, E)be a simple undirected graph modeling friendships among six people: V=v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , E= v 1 , v 2 ,v 1 , v 3 ,v 2 , v 3 ,v 2 , v 4 ,v 3 , v 5 ,v 4 , v 5 ,v 5 , v 6 . LetCbe a set of parameters describing interaction contexts: C=c Work , c Sport , c Online . 79 Chapter 4. Applications of Soft Set DefineA:C→P(V)(active people) andB:C→P(E)(active friendships) by: A(c Work ) =v 1 , v 2 , v 3 , v 4 ,B(c Work ) =v 1 , v 2 ,v 1 , v 3 ,v 2 , v 3 ,v 2 , v 4 , A(c Sport ) =v 2 , v 3 , v 5 , v 6 ,B(c Sport ) =v 2 , v 3 ,v 3 , v 5 ,v 5 , v 6 , A(c Online ) =v 1 , v 3 , v 5 ,B(c Online ) =v 1 , v 3 ,v 3 , v 5 . For eachc∈C, every edge inB(c)has both endpoints inA(c), hence B(c)⊆ u, v∈E:u, v∈A(c) . Therefore, ( G, C, A, B ) is a soft graph overG. Interpretation: the underlying friendship network isG, while each parametercextracts a context- dependent subgraph (e.g., work interactions, sports interactions, online interactions). 4.2 Soft Topological Space A soft topological space is a parameterized family of soft open sets closed under soft unions and finite intersections, containing the null and absolute soft sets [227,228]. Related notions include hypersoft topological spaces [229–231], fuzzy soft topological spaces [232,233], intuitionistic fuzzy soft topological spaces [234,235], and neutrosophic soft topological spaces [227,236–238]. Definition 4.2.1(Soft topology and soft topological space).[227,228] LetXbe a nonempty universe and letAbe a nonempty set of parameters. Asoft setoverX(with parameter setA) is a pair(F, A)whereF:A→P(X). Denote bySS(X, A)the collection of all such soft sets. Define theA-nullandA-absolutesoft sets by 0 A = (F 0 , A), F 0 (a) =∅(a∈A),1 A = (F 1 , A), F 1 (a) =X(a∈A). For a family(F i , A) i∈I ⊆S(X, A), define the soft union ⊔ i∈I (F i , A) = (F, A)by F(a) = ⋃ i∈I F i (a) (a∈A), and for(F, A),(G, A)∈S(X, A)define the soft intersection(F, A)u(G, A) = (H, A)by H(a) =F(a)∩G(a) (a∈A). A subfamilyτ⊆S(X, A)is called asoft topologyonX(with parameter setA) if (ST1)0 A ,1 A ∈τ; (ST2) if(F, A),(G, A)∈τ, then(F, A)u(G, A)∈τ; (ST3) if(F i , A)∈τfor alli∈I, then ⊔ i∈I (F i , A)∈τ. 80 Chapter 4. Applications of Soft Set In this case, the triple(X, τ, A)is called asoft topological space. Elements ofτare calledsoft open sets; a soft set(F, A)issoft closedif its soft complement(F, A) c belongs toτ. Example 4.2.2(Example of a soft topological space: neighborhood-based accessibility under different criteria).Let X=x 1 , x 2 , x 3 be a set of locations (e.g., three service points), and let A=a Walk , a Drive be a set of parameters, wherea Walk means “reachable on foot” anda Drive means “reachable by car”. Define the following soft sets overX(all with parameter setA): 0 A (a Walk ) = 0 A (a Drive ) =∅,1 A (a Walk ) = 1 A (a Drive ) =X, and a nontrivial soft set(F, A)by F(a Walk ) =x 1 , x 2 ,F(a Drive ) =x 1 , x 2 , x 3 . Now define τ=0 A ,1 A ,(F, A) ⊆S(X, A). We verify thatτis a soft topology onX: (ST1)0 A ,1 A ∈τby definition. (ST2) Finite soft intersections remain inτ: (F, A)u1 A = (F, A),(F, A)u0 A = 0 A ,(F, A)u(F, A) = (F, A). (ST3) Arbitrary soft unions of members ofτremain inτ: 0 A ⊔ (F, A) = (F, A),(F, A) ⊔ 1 A = 1 A ,0 A ⊔ 1 A = 1 A , and any union of copies of(F, A)is still(F, A). Hence(X, τ, A)is a soft topological space. Interpretation: under the walking criterion, the “open” accessible region isx 1 , x 2 , while under driving it isX; the soft topologyτcontains the null region, the whole region, and this parameter- dependent accessibility region. 81 Chapter 4. Applications of Soft Set 4.3 Soft Algebra A soft algebra is a family of soft sets closed under soft complement and finite soft unions, containing the null soft set [239–242]. Definition 4.3.1(Soft algebra).[239, 240] LetXbe a nonempty universe and letQbe a nonempty set of parameters. Asoft setoverX(with parameter setQ) is a pair(F, Q)where F:Q−→P(X). Denote byS Q (X)the family of all such soft sets. Define thenull soft setΦ Q = (F 0 , Q)and theabsolute soft setX Q = (F 1 , Q)by F 0 (q) =∅,F 1 (q) =X(q∈Q). For(F, Q)∈S Q (X)define thesoft complement(F, Q) c = (F c , Q)by F c (q) =X (q) (q∈Q). For(F, Q),(G, Q)∈S Q (X)define thesoft unionandsoft intersectionby (F, Q) ̃ ∪(G, Q) = (H, Q), H(q) =F(q)∪G(q),(F, Q) ̃ ∩(G, Q) = (K, Q), K(q) =F(q)∩G(q). A subfamilyΣ⊆S Q (X)is called asoft algebraonX(parameterized byQ) if: (SA1)Φ Q ∈Σ; (SA2) if(F, Q)∈Σ, then(F, Q) c ∈Σ; (SA3) if(F i , Q)∈Σfori= 1,2, . . . , k, then ̃ ⋃ k i=1 (F i , Q)∈Σ, equivalently,Σis closed under finite soft unions. Example 4.3.2(Example of a soft algebra: “fixed-or-empty” soft sets).Let X=1,2,3,4andQ=q 1 , q 2 be a universe and a parameter set. Fix two subsets ofX: A=1,2,B=3,4. Consider the following subfamilyΣ⊆S Q (X)consisting of all soft sets(F, Q)such that for each parameterq∈Qone has F(q)∈∅, A, B, X. Equivalently, Σ = (F, Q)∈S Q (X) ∣ ∣ ∣ F(q 1 ), F(q 2 )∈∅, A, B, X . ThenΣis a soft algebra onX(parameterized byQ): 82 Chapter 4. Applications of Soft Set (SA1) The null soft setΦ Q belongs toΣsinceΦ Q (q 1 ) = Φ Q (q 2 ) =∅. (SA2) If(F, Q)∈Σ, then for eachq∈Qwe haveF(q)∈∅, A, B, X, and hence F c (q) =X (q)∈X, B, A,∅⊆∅, A, B, X. Thus(F, Q) c ∈Σ. (SA3) If(F 1 , Q), . . . ,(F k , Q)∈Σ, then for eachq∈QeveryF i (q)is one of∅, A, B, X, and therefore k ⋃ i=1 F i (q)∈∅, A, B, X. Hence ̃ ⋃ k i=1 (F i , Q)∈Σ. Interpretation:Σmodels a simplified decision system where, for each parameter, only four outcomes are allowed (select nothing, select groupA, select groupB, or select allX), and the family remains stable under combining rules (union) and negating rules (complement). 4.4 Soft Lattice A soft lattice is a lattice structure on soft sets (often modulo soft equality), using soft union and intersection operations [243–245]. Related notions include fuzzy soft lattices [246, 247], intuitionistic fuzzy soft lattices [248–250], and neutrosophic soft lattices [243]. Definition 4.4.1(Soft lattice).[243–245] LetUbe a nonempty universe and letEbe a nonempty set of parameters. Write S(U, E) :=(F, A)|A⊆E, F:A→P(U) for the collection of all (crisp) soft sets overU. (1) Operations.For(F, A),(G, B)∈S(U, E)define theextended union (F, A) ̃ ∪(G, B) := (H, A∪B), where for eache∈A∪B, H(e) = F(e),e∈A , G(e),e∈B , F(e)∪G(e), e∈A∩B, and define therestricted intersection (F, A) ̃ ∩(G, B) := (K, A∩B),K(e) =F(e)∩G(e) (e∈A∩B). (2) Generalized (1-)soft equality.Define a binary relation 1 onS(U, E)by (F, A) 1 (G, B) :⇐⇒A=∅or ( ∀e∈A∃e ′ ∈B:F(e)⊆G(e ′ ) ) . 83 Chapter 4. Applications of Soft Set Define1-soft equality≈ 1 by (F, A)≈ 1 (G, B) :⇐⇒(F, A) 1 (G, B)and(G, B) 1 (F, A). Then≈ 1 is an equivalence relation onS(U, E). (3) The soft lattice (quotient lattice).LetS(U, E)/≈ 1 be the set of≈ 1 -equivalence classes, and write[(F, A)]for the class of(F, A). Define binary operations∨,∧onS(U, E)/≈ 1 by [(F, A)]∨[(G, B)] := [ (F, A) ̃ ∪(G, B) ] ,[(F, A)]∧[(G, B)] := [ (F, A) ̃ ∩(G, B) ] . Asoft lattice(with respect to≈ 1 ) is the lattice ( S(U, E)/≈ 1 ,∨,∧ ) , which satisfies the lattice axioms (commutativity, associativity, idempotency, and absorption) in the usual sense on equivalence classes. If one additionally specifies the bottom and top elements (1-null and 1-universal soft sets), then it becomes a bounded soft lattice. 4.5 Soft Vector A soft vector is a parameter-indexed mapping into a vector space, representing context-dependent vectors and enabling membership in soft sets [251]. Definition 4.5.1(Soft vector (soft element of a vector space)).[251] LetVbe a vector space over a fieldF, and letAbe a nonempty set of parameters. Asoft vectoroverV(with respect toA) is a mapping ̃v:A−→V. If(F, A)is a soft set overV, i.e.F:A→P(V), then a soft vector ̃vis said tobelongto(F, A) (denoted ̃v∈(F, A)) if ̃v(a)∈F(a)for alla∈A. Remark 4.5.2.A soft vector ̃vis calledconstantif there existsv∈Vsuch that ̃v(a) =vfor alla∈A. Example 4.5.3(Example of a soft vector: portfolio weights under different market scenarios). LetV=R 3 be the vector space of portfolio weight vectors for three assets (Asset 1, Asset 2, Asset 3) over the fieldR. Let the parameter set represent market scenarios: A=a Bull , a Base , a Bear . Define a soft vector ̃v:A→Vby assigning a (scenario-dependent) weight vector to each scenario: ̃v(a Bull ) = (0.50,0.30,0.20), ̃v(a Base ) = (0.40,0.40,0.20), ̃v(a Bear ) = (0.20,0.50,0.30). 84 Chapter 4. Applications of Soft Set Thus ̃vis a soft vector overV. Now define a soft set(F, A)overVdescribing admissible portfolios under each scenario: F(a Bull ) =(x 1 , x 2 , x 3 )∈R 3 |x 1 ≥0.40, x 2 ≤0.40, x 3 ≤0.30, F(a Base ) =(x 1 , x 2 , x 3 )∈R 3 |0.30≤x 1 ≤0.50,0.30≤x 2 ≤0.50, x 3 = 0.20, F(a Bear ) =(x 1 , x 2 , x 3 )∈R 3 |x 1 ≤0.30, x 2 ≥0.40, x 3 ≥0.20. Then ̃v∈(F, A)because, for everya∈A, the vector ̃v(a)satisfies the corresponding constraints and hence belongs toF(a). Interpretation: the soft vector encodes a portfolio recommendation that adapts to market scenar- ios, and membership in the soft set expresses feasibility of the recommendation under scenario- specific rules. 4.6 Soft functions A soft function maps soft sets between universes via underlying object and parameter maps, transporting approximations through images and preimages [252–254]. Definition 4.6.1(Soft function (soft mapping between soft classes)).LetXandYbe nonempty universes, and letAandBbe nonempty parameter sets. Denote S(X, A) :=(F, A)|F:A→P(X),S(Y, B) :=(G, B)|G:B→P(Y). Letu:X→Yandp:A→Bbe mappings. Thesoft function induced by(u, p)is the mapping f pu :S(X, A)−→S(Y, B), defined as follows. (1) Image. For(F, A)∈S(X, A), define its image f pu (F, A) := (F ′ , p(A)), whereF ′ :B→P(Y)is given, for eachb∈B, by F ′ (b) = ⋃ a∈p −1 (b)∩A u ( F(a) ) , p −1 (b)∩A6=∅, ∅,otherwise. (Hereu(F(a)) =u(x)|x∈F(a)⊆Y.) (2) Inverse image. For(G, B)∈S(Y, B), define its inverse image f −1 pu (G, B) := (H, p −1 (B)), whereH:A→P(X)is given, for eacha∈A, by H(a) = u −1 ( G(p(a)) ) , p(a)∈B, ∅,otherwise. The soft functionf pu is calledsoft injective(resp.soft surjective) if bothuandpare injective (resp. surjective). 85 Chapter 4. Applications of Soft Set Example 4.6.2(Example of a soft function induced by(u, p)).Let X=x 1 , x 2 , x 3 , x 4 andY=y 1 , y 2 , y 3 be universes, and let A=a 1 , a 2 , a 3 ,B=b 1 , b 2 be parameter sets. Define an object-mappingu:X→Yand a parameter-mappingp:A→Bby u(x 1 ) =y 1 , u(x 2 ) =y 1 , u(x 3 ) =y 2 , u(x 4 ) =y 3 , p(a 1 ) =b 1 , p(a 2 ) =b 1 , p(a 3 ) =b 2 . Consider the soft set(F, A)∈S(X, A)defined by F(a 1 ) =x 1 , x 3 ,F(a 2 ) =x 2 ,F(a 3 ) =x 4 . We compute its image under the soft functionf pu . Sincep(A) =b 1 , b 2 , we havef pu (F, A) = (F ′ , p(A))withF ′ :B→P(Y)given by F ′ (b 1 ) =u(F(a 1 ))∪u(F(a 2 )) =u(x 1 ), u(x 3 )∪u(x 2 )=y 1 , y 2 , F ′ (b 2 ) =u(F(a 3 )) =u(x 4 )=y 3 . Hence f pu (F, A) = (F ′ ,b 1 , b 2 ),F ′ (b 1 ) =y 1 , y 2 , F ′ (b 2 ) =y 3 . Next, let(G, B)∈S(Y, B)be given by G(b 1 ) =y 1 ,G(b 2 ) =y 2 , y 3 . Its inverse image underf pu isf −1 pu (G, B) = (H, p −1 (B)) = (H, A), where H(a 1 ) =u −1 (G(p(a 1 ))) =u −1 (G(b 1 )) =u −1 (y 1 ) =x 1 , x 2 , H(a 2 ) =u −1 (G(p(a 2 ))) =u −1 (G(b 1 )) =x 1 , x 2 , H(a 3 ) =u −1 (G(p(a 3 ))) =u −1 (G(b 2 )) =u −1 (y 2 , y 3 ) =x 3 , x 4 . Thusf pu transports soft information from(X, A)to(Y, B)by combining parameters viapand pushing forward object-sets viau, whilef −1 pu pulls soft information back by preimages. Remark 4.6.3(A common special case: fixed parameters).IfA=Bandp=id A , thenf pu reduces to the soft mapping induced only byu: f u (F, A) = (F u , A),F u (a) =u(F(a)) (a∈A). 86 Chapter 4. Applications of Soft Set 4.7 Soft groups A soft group assigns each parameter a subgroup of a given group, forming a parameterized family of subgroups [255,256]. Related notions include hypersoft groups [257], fuzzy soft groups [258], and neutrosophic soft groups [259]. Definition 4.7.1(Soft group).[255,256] LetGbe a group with identity elemente, and letE be a nonempty set of parameters. LetA⊆Ebe nonempty. Asoft setoverGis a pair(F, A) where F:A−→P(G). The soft set(F, A)is called asoft group overGif F(a)≤Gfor alla∈A, i.e., for every parametera, the valueF(a)is a (classical) subgroup ofG. Equivalently, a soft group is a parameterized family of subgroups ofG. We may also denote a soft group by the triple(G, F, A). Remark 4.7.2(Standard special cases).Let(F, A)be a soft group overG. (i)(F, A)is anidentity soft groupifF(a) =efor alla∈A. (i)(F, A)is anabsolute soft groupifF(a) =Gfor alla∈A. Definition 4.7.3(Soft subgroup).Let(F, A)and(H, B)be soft groups over the same group G. We say that(H, B)is asoft subgroupof(F, A), and write(H, B) ̃ ≤(F, A), if B⊆AandH(b)≤F(b)for allb∈B. Example 4.7.4(Example of a soft subgroup).LetG= (Z,+)be the additive group of integers, and let E=a 2 , a 4 , a 6 be a set of parameters. TakeA=E. Define a soft group(F, A)overGby assigning, to each parameter, a subgroup ofZ: F(a 2 ) = 2Z,F(a 4 ) = 4Z,F(a 6 ) = 6Z. EachF(a i )is a subgroup of(Z,+), hence(F, A)is a soft group. Now take the subsetB=a 4 , a 6 ⊆Aand define another soft group(H, B)by H(a 4 ) = 8Z,H(a 6 ) = 12Z. Again,H(a 4 )andH(a 6 )are subgroups ofZ, so(H, B)is a soft group. Moreover, for eachb∈Bwe have subgroup inclusions H(a 4 ) = 8Z≤4Z=F(a 4 ),H(a 6 ) = 12Z≤6Z=F(a 6 ). Since alsoB⊆A, it follows that (H, B) ̃ ≤(F, A), i.e.,(H, B)is a soft subgroup of(F, A). 87 Chapter 4. Applications of Soft Set 4.8 Soft Field A soft field assigns each parameter a subfield of a fixed field, forming a parameterized family closed under field operations. Definition 4.8.1(Soft field).Let(K,+,·)be a (crisp) field, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. Asoft field overK(with parameter setA) is a soft set(F, A)overK, i.e., F:A−→P(K), such that for everya∈A, the setF(a)⊆Kis a (classical) subfield ofK. Equivalently, for each a∈Athe following conditions hold: (i)0,1∈F(a)and16 = 0; (i) ifx, y∈F(a), thenx−y∈F(a); (i) ifx, y∈F(a), thenx·y∈F(a); (iv) ifx∈F(a)andx6= 0, thenx −1 ∈F(a). Thus, a soft field is a parameterized family of subfields of the fixed ground fieldK. Remark 4.8.2(Related notions).(i) Replacing “subfield” by “subring” yields the standard notion of a soft ring. (i) In uncertainty-aware extensions, one replaces “subfield” by an appropriate uncertain ana- logue (e.g., aneutrosophic subfield), obtaining a neutrosophic soft field. (i) Some authors also define “soft field” via thesoft-elementframework as a commutative soft ring with soft unity in which every nonzero soft element is a soft unit. 4.9 Soft Ring A soft ring assigns each parameter a subring of a fixed ring, forming a parameterized family closed under addition and multiplication [260–262]. Definition 4.9.1(Soft ring).Let(R,+,·)be a (not necessarily unital) ring, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. A pair(F, A)is called asoft ring overRif F:A−→P(R) is a mapping such that, for everya∈A, the setF(a)⊆Ris a (classical) subring ofR. Equivalently, for eacha∈A: F(a)6 =∅,x−y∈F(a)andx·y∈F(a) (∀x, y∈F(a)). Thus a soft ring is a parameterized family of subrings of the fixed ground ringR. 88 Chapter 4. Applications of Soft Set Remark 4.9.2.If, additionally, eachF(a)is an ideal ofR(instead of merely a subring), then (F, A)is called asoft idealoverR. Example 4.9.3(Example of a soft ring).LetR= (Z,+,·)be the ring of integers, and let E=a 2 , a 3 , a 6 be a parameter set. TakeA=E. Define a mappingF:A→P(Z)by F(a 2 ) = 2Z,F(a 3 ) = 3Z,F(a 6 ) = 6Z, wherenZ:=nk|k∈Z. EachF(a i )is a nonempty subring ofZ: ifx=nkandy=n`belong tonZ, then x−y=n(k−`)∈nZ,x·y=n 2 k`∈nZ. Therefore(F, A)is a soft ring overZ. Moreover, eachnZis actually an ideal ofZ, so(F, A)is also a soft ideal in the sense of Re- mark 4.9.2. 4.10 Soft Matroid A matroid is a combinatorial independence structure generalizing linear independence, defined via independent sets satisfying hereditary and exchange axioms [263–265]. A soft matroid is a parameterized matroid-like structure on soft sets, using soft points and exchange axioms for independence [266,267]. Definition 4.10.1(Soft-matroid).LetUbe a universal set,Ea set of parameters, andA⊆E nonempty. LetF A = (F, A)be afinitesoft set overU, i.e.,F:A→P(U)andF(e)is finite for alle∈A. Asoft-pointofF A is a soft setp e x = (P, A)such thatP(e) =xfor somee∈AandP(e ′ ) =∅ for alle ′ ∈A\e; we writep e x ̃ ∈F A wheneverx∈F(e). Thecardinality|F A |is the number of soft-points belonging toF A . Let ̃ ⊆denote the soft subset relation on soft sets with (possibly) different parameter domains, and let ̃ ∪and ̃ the usual soft union and soft difference (defined parameterwise). Let∅ A be the null soft set onA(i.e.,F(e) =∅for alle∈A). Asoft-matroidis an ordered pair ̃ M= (F A ,G), whereGis a collection of sub-soft-sets ofF A satisfying the following axioms: 89 Chapter 4. Applications of Soft Set (SM1)(Null axiom)∅ A ∈G. (SM2)(Hereditary axiom)IfG A ∈GandG ′ A ̃ ⊆G A , thenG ′ A ∈G. (SM3)(Exchange axiom)IfG A , H A ∈Gwith|G A |<|H A |, then there exists a soft-pointp e x ofH A ̃ A such that G A ̃ ∪p e x ∈G. In this case, ̃ Mis called a soft-matroid onF A . Example 4.10.2(Example of a soft-matroid: selecting nonredundant skills across job roles). LetUbe a finite set of skills: U=σ 1 , σ 2 , σ 3 , σ 4 , whereσ 1 = Python,σ 2 = Databases,σ 3 = Cloud,σ 4 = Security. LetA=e 1 , e 2 , e 3 ⊆Ebe a set of parameters (job roles), wheree 1 = Backend,e 2 = Data,e 3 = DevOps. Define a finite soft setF A = (F, A)overUby F(e 1 ) =σ 1 , σ 2 ,F(e 2 ) =σ 1 , σ 2 , σ 3 ,F(e 3 ) =σ 1 , σ 3 , σ 4 . A soft-pointp e σ belongs toF A (i.e.,p e σ ̃ ∈F A ) exactly whenσ∈F(e). LetGbe the family of all sub-soft-setsG A = (G, A)ofF A satisfying the followingat most one skill per rolerule: ∣ ∣ G(e) ∣ ∣ ≤1for everye∈A. (Thus, for each rolee, eitherG(e) =∅orG(e) =σfor someσ∈F(e).) Then ̃ M= (F A ,G)is a soft-matroid: (i)Null axiom:the null soft set∅ A satisfies|∅ A (e)|= 0≤1for alle, hence∅ A ∈G. (i)Hereditary axiom:ifG A ∈GandG ′ A ̃ ⊆G A , then|G ′ (e)|≤|G(e)|≤1for alle, soG ′ A ∈G. (i)Exchange axiom:letG A , H A ∈Gwith|G A |<|H A |(i.e.,H A has more soft-points). Then there exists some rolee ∗ ∈Asuch thatH(e ∗ )6 =∅andG(e ∗ ) =∅. Choose the uniqueσ ∗ withH(e ∗ ) =σ ∗ , and consider the soft-pointp e ∗ σ ∗ . It is a soft-point ofH A ̃ A , and adding it toG A gives (G A ̃ ∪p e ∗ σ ∗ )(e ∗ ) =σ ∗ ,(G A ̃ ∪p e ∗ σ ∗ )(e) =G(e) (e6 =e ∗ ), so the “at most one skill per role” condition still holds. HenceG A ̃ ∪p e ∗ σ ∗ ∈G. Interpretation:Grepresents feasible (independent) selections ofnonredundantskills across roles, and the exchange axiom formalizes the ability to extend a smaller feasible assignment by adding a skill from a larger feasible assignment in a role where nothing was chosen yet. 90 Chapter 4. Applications of Soft Set 4.11 Soft Bitopological Space A soft bitopological space equips a universe with two soft topologies, supporting dual soft open- ness and separation analysis simultaneously [268–271]. Definition 4.11.1(Soft bitopological space).LetXbe a nonempty set and letAbe a nonempty set of parameters. Letτ 1 andτ 2 be two (not necessarily equal) soft topologies onXwith respect to the same parameter setA. Then the quadruple (X, A, τ 1 , τ 2 ) is called asoft bitopological space. A soft set(F, A)∈τ i is calledτ i -soft open(i= 1,2), and (F, A)isτ i -soft closedif its soft complement(F, A) c belongs toτ i . Example 4.11.2(A soft bitopological space for two “notions of openness” in urban accessibility). LetXbe a finite set of city districts (alternatives) X=x 1 , x 2 , x 3 , x 4 , and let the parameter set be A=Transit,Safety. Intuitively, we will model two different criteria for when a family of districts is regarded as “soft open”: one based onpublic-transport accessibilityand the other based onpublic safety. Define the following soft sets overX(with parameter setA): 0 A (a) =∅,1 A (a) =X(a∈A), and two nontrivial soft sets(F, A)and(G, A)by F(Transit) =x 1 , x 2 , F(Safety) =x 1 , x 3 , G(Transit) =x 2 , x 4 , G(Safety) =x 3 , x 4 . Soft topologyτ 1 (Transit-openness).Let τ 1 :=0 A ,1 A ,(F, A),(F, A) c . Thenτ 1 is a soft topology onX(with parameter setA), since it contains0 A ,1 A , is closed under finite soft intersections, and under arbitrary soft unions. Soft topologyτ 2 (Safety-openness).Let τ 2 :=0 A ,1 A ,(G, A),(G, A) c . Similarly,τ 2 is a soft topology onX. Therefore the quadruple (X, A, τ 1 , τ 2 ) is a soft bitopological space in the sense of Definition 4.11.1. Interpretation.Aτ 1 -soft open set represents districts that are “open” under the transit-based viewpoint, while aτ 2 -soft open set represents districts that are “open” under the safety-based viewpoint. For instance,(F, A)∈τ 1 encodes a transit-favorable selection (parameterwise), whereas(G, A)∈τ 2 encodes a safety-favorable selection. 91 Chapter 4. Applications of Soft Set 4.12 Soft Module A soft module is a parameterized family of submodules of a fixed module, enabling context- dependent linear structure modeling [272–276]. Definition 4.12.1(Soft module).LetRbe a ring, letMbe a (left)R-module, and letAbe a nonempty set of parameters. Asoft setoverMwith parameter setAis a pair(F, A)where F:A−→P(M). The soft set(F, A)is called asoft module overMif, for everya∈A, the valueF(a)is a (classical)R-submodule ofM, i.e., F(a)≤M(a∈A). Equivalently, for eacha∈A: 0 M ∈F(a),x−y∈F(a)andrx∈F(a) (∀x, y∈F(a),∀r∈R). Thus, a soft module is a parameterized family of submodules of the fixed ground moduleM. Example 4.12.2(A soft module for permission-dependent access subspaces).LetR=Rand letM=R 3 be the standard leftR-module. Interpret vectorsm= (m 1 , m 2 , m 3 )∈Masfeature triples(e.g., three measurable attributes of a record). Let the parameter set be A=public,internal,admin, representing three access policies. Define a mappingF:A→P(M)by F(public) =(x,0,0) :x∈R, F(internal) =(x, y,0) :x, y∈R, F(admin) =R 3 . Then eachF(a)is anR-submodule ofM(indeed, a linear subspace): it contains0 M = (0,0,0), is closed under subtraction, and is closed under scalar multiplication. Hence(F, A)is a soft module overMin the sense of Definition 4.12.1. Interpretation.Underpublicaccess, only the first coordinate can vary (others must be hidden as0);internalaccess reveals the first two coordinates; andadminaccess reveals all three. Thus the admissible information for each policy forms a submodule, and the collection of these submodules indexed byAis captured by the soft module(F, A). 92 Chapter 4. Applications of Soft Set 4.13 Soft Metric Space A soft metric space assigns each parameter a metric on the universe, measuring distances under varying contexts and uncertainty scenarios [277–280]. Definition 4.13.1(Soft point and soft element).LetXbe a nonempty universe and letEbe a nonempty set of parameters. (i) Asoft pointofXis a soft setP e x = (P, E)such that P(e) =xandP(e ′ ) =∅(e ′ ∈E\e), for some(e, x)∈E×X. (i) Asoft elementofX(with supportA⊆E) is a soft setα A = (α, A)overXsuch that for everya∈A, α(a) =x a for somex a ∈X,andα(e) =∅(e∈E ). Equivalently, a soft element is a (partial) choice functionA→X, a7→x a . We write SE(X, E)for the collection of all soft elements ofXwhose supports are contained inE. Definition 4.13.2(Soft real numbers and order).Asoft real number(overE) is a mapping ̃r:E→R. It isnonnegativeif ̃r(e)≥0for alle∈E. Denote by ̃ R E + the set of all nonnegative soft real numbers. For ̃r, ̃s∈ ̃ R E + define: ( ̃r⊕ ̃s)(e) := ̃r(e) + ̃s(e), ̃r≤ ̃s⇐⇒ ̃r(e)≤ ̃s(e) (∀e∈E). Let ̃ 0∈ ̃ R E + be the zero soft real number, ̃ 0(e) = 0for alle∈E. Definition 4.13.3(Soft metric and soft metric space).LetXbe a nonempty universe and let Ebe a nonempty parameter set. A mapping d S :SE(X, E)×SE(X, E)−→ ̃ R E + is called asoft metric(orsoft distance) on(X, E)if, for allα, β, γ∈SE(X, E), the following axioms hold: (SM1)(Nonnegativity) ̃ 0≤d S (α, β). (SM2)(Identity of indiscernibles)d S (α, β) = ̃ 0if and only ifα=β. (SM3)(Symmetry)d S (α, β) =d S (β, α). 93 Chapter 4. Applications of Soft Set (SM4)(Triangle inequality)d S (α, γ)≤d S (α, β)⊕d S (β, γ). In this case, the triple(X, E, d S )is called asoft metric space. Remark 4.13.4(Relation to the soft-point (product) viewpoint).In the original approach of Das–Samanta, a “soft metric” is defined on the set of all soft points, which can be identified withE×X; hence it essentially reduces to an ordinary metric on the product setE×X. The soft-element based definition above generalizes that viewpoint: every soft-point metric induces a soft-element metric, but the converse need not hold. Example 4.13.5(A soft metric space for multi-context sensor readings).LetX=Rbe the set of possible temperature readings, and let E=morning,noon,night be a parameter set representing three measurement contexts (time slots). A soft elementα∈SE(X, E)assigns to eache∈Ea singletonα(e) =x e , so we identifyα with the triple(x morning , x noon , x night )∈R 3 . Defined S :SE(X, E)×SE(X, E)→ ̃ R E + by, forα= (x e ) e∈E andβ= (y e ) e∈E , d S (α, β)(e) := ∣ ∣ x e −y e ∣ ∣ (e∈E). Equivalently,d S (α, β)is the soft real numbere7→|x e −y e |. Thend S is a soft metric: •Nonnegativity:d S (α, β)(e) =|x e −y e |≥0for alle, hence ̃ 0≤d S (α, β). •Identity:d S (α, β) = ̃ 0iff|x e −y e |= 0for alle, i.e.x e =y e for alle, henceα=β. •Symmetry:|x e −y e |=|y e −x e |givesd S (α, β) =d S (β, α). •Triangle inequality:for eache∈E, d S (α, γ)(e) =|x e −z e |≤|x e −y e |+|y e −z e |=d S (α, β)(e)+d S (β, γ)(e) = ( d S (α, β)⊕d S (β, γ) ) (e), henced S (α, γ)≤d S (α, β)⊕d S (β, γ). Therefore(X, E, d S )is a soft metric space. Interpretation.Each soft element represents a day’s temperature profile across contexts (morn- ing/noon/night), and the soft distance returns the absolute discrepancy at each context as a nonnegative soft real number. 94 Chapter 4. Applications of Soft Set 4.14 Soft probabilities Soft probabilities assign to each parameter a probability measure on a universe, representing context-dependent stochastic uncertainty for decision-making tasks explicitly [281–283]. Definition 4.14.1(Statistical database, admissible samples, and frequency).LetΩbe a nonempty outcome space. Astatistical database(of lengthN∈N) is a finite sequence Base= (ω 1 , ω 2 , . . . , ω N ),ω i ∈Ω. Fix an integermwith1≤m≤Nand a “freshness” indexτwith1≤τ≤N−m+ 1. Define the family ofadmissible samples(consecutive blocks) by S(Base, m, τ) := i, i+ 1, . . . , i+m−1 ∣ ∣ i=τ, τ+ 1, . . . , N−m+ 1 . For an eventA⊆Ωand a sampleI∈S(Base, m, τ), itsfrequency of occurrenceonIis μ(Base, A, I) := 1 |I| ∑ i∈I χ A (ω i ),χ A (ω) = 1, ω∈A, 0, ω/∈A. Definition 4.14.2(Soft probability (interval-valued, parameterized by(m, τ))).Let Base,m, andτbe as above. Thesoft probabilityof an eventA⊆Ωon the databaseBaseat parameters (m, τ)is the closed interval P soft (A|Base;m, τ) := [ p (A),p(A) ] ⊆[0,1], where p (A) :=min I∈S(Base,m,τ) μ(Base, A, I),p(A) :=max I∈S(Base,m,τ) μ(Base, A, I). Equivalently, P soft (A|Base;m, τ) =λ ( χ A ,Base, m, τ ) , whereλ(f,Base, m, τ)denotes the(m, τ)-approximate mean intervalof a real-valued function f: Ω→R: λ(f,Base, m, τ) := [ min I∈S(Base,m,τ) 1 |I| ∑ i∈I f(ω i ),max I∈S(Base,m,τ) 1 |I| ∑ i∈I f(ω i ) ] . Example 4.14.3(Soft probability for a delayed-train event under rolling windows).LetΩ = ω 1 , . . . , ω 10 be ten commuting days, and let Base= (ω i , y i ) 10 i=1 be a database wherey i ∈0,1indicates whether the train was delayed on dayω i (y i = 1means “delayed”). Consider the event A:=ω i ∈Ω :y i = 1. Assume the observed delay indicators are (y 1 , . . . , y 10 ) = (1,0,1,0,0,1,1,0,0,1). 95 Chapter 4. Applications of Soft Set Fix parameters(m, τ)so that the admissible index-family is the set of all contiguous windows of lengthm= 4(this is a concrete choice ofS(Base, m, τ)): S(Base,4, τ) := 1,2,3,4,2,3,4,5, . . . ,7,8,9,10 . For each windowI∈S(Base,4, τ)define μ(Base, A, I) := 1 |I| ∑ i∈I χ A (ω i ) = 1 4 ∑ i∈I y i , i.e. the empirical delay rate inside that window. Compute the window means: I (y i ) i∈I μ(Base, A, I) 1,2,3,4(1,0,1,0)2/4 = 0.50 2,3,4,5(0,1,0,0)1/4 = 0.25 3,4,5,6(1,0,0,1)2/4 = 0.50 4,5,6,7 (0,0,1,1)2/4 = 0.50 5,6,7,8(0,1,1,0)2/4 = 0.50 6,7,8,9(1,1,0,0)2/4 = 0.50 7,8,9,10(1,0,0,1)2/4 = 0.50 Hence p (A) =min I∈S(Base,4,τ) μ(Base, A, I) = 0.25,p(A) =max I∈S(Base,4,τ) μ(Base, A, I) = 0.50. Therefore the soft probability of “delay” on the database Base at parameters(4, τ)is P soft (A|Base; 4, τ) = [0.25,0.50]. Interpretation.Depending on which admissible 4-day period (window) is regarded as represen- tative under(m, τ), the estimated delay probability ranges from25% to50%, so the uncertainty is captured as an interval. Remark 4.14.4(Basic sanity properties).For fixed(Base, m, τ): 1.P soft (∅|Base;m, τ) = [0,0]andP soft (Ω|Base;m, τ) = [1,1]. 2. IfA⊆B, thenP soft (A|Base;m, τ)⊆P soft (B|Base;m, τ)in the endpoint-wise order (monotonicity). 3. Ifp (A) =p(A), the soft probability ofAcollapses to a classical (database-induced) point probability. 96 Chapter 4. Applications of Soft Set 4.15 Soft SemiGroup A semigroup is a nonempty set equipped with an associative binary operation, requiring closure and no identity or inverses [98, 284, 285]. A soft semigroup is a soft set whose value at each parameter is a subsemigroup of a fixed semigroup, closed under multiplication [286–289]. Definition 4.15.1(Soft semigroup).Let(S,·)be a semigroup, letEbe a nonempty set of parameters, and letA⊆Ebe nonempty. Asoft semigroup overSis a soft set(F, A)overS, i.e., F:A−→P(S), such that for every parametera∈A, the value setF(a)is a (classical) subsemigroup ofS. Equivalently, for eacha∈A: F(a)6=∅andx·y∈F(a)for allx, y∈F(a). Thus, a soft semigroup is a parameterized family of subsemigroups of the fixed ground semigroup S. Remark 4.15.2.IfSis equipped with a compatible partial order≤(so(S,·,≤)is anordered semigroup), then(F, A)is often called asoft ordered semigroupwhen eachF(a)is a subsemigroup ofS(for alla∈A). Example 4.15.3(A soft semigroup on(N 0 ,+)).Let(S,·) = (N 0 ,+)be the additive semigroup of nonnegative integers. Let the parameter set be E=Even,Mult3,AtLeast5,A=E. Define a soft set(F, A)overSby F(Even) =0,2,4,6, . . .,F(Mult3) =0,3,6,9, . . .,F(AtLeast5) =5,6,7, . . .. Then for eacha∈A, the valueF(a)is a nonempty subsemigroup of(N 0 ,+): • Ifx, y∈F(Even), thenx+yis even, sox+y∈F(Even). • Ifx, y∈F(Mult3), thenx+yis a multiple of3, sox+y∈F(Mult3). • Ifx, y∈F(AtLeast5), thenx+y≥10, hencex+y∈F(AtLeast5). Therefore(F, A)is a soft semigroup overSin the sense of Definition 4.15.1. Real-life interpretation.LetSrepresent feasible daily production counts in a factory. The parameterEvenenforces pairing/packaging constraints (even batch sizes),Mult3models palleti- zation in groups of three, andAtLeast5encodes a minimum-run policy; each policy set is closed under combining runs. 97 Chapter 4. Applications of Soft Set 4.16 Soft HyperStructure and SuperHyperStructure A hyperstructure is an algebraic system whose binary operation assigns each element-pair a nonempty subset, generalizing ordinary operations [290,291]. A superhyperstructure iterates hy- perstructural levels so hyperoperations act on set-valued objects across multiple power-set layers, forming hierarchies [292]. A soft hyperstructure is a parameterized family of subhyperstructures, assigning each parameter a nonempty subset closed under the underlying hyperoperation(s) of a given hyperalgebra [293–295]. Definition 4.16.1(Soft HyperStructure).[293–295] LetHbe a hyperalgebra (hyperstructure) with signature Σ =? i :H n i →P ∗ (H)|i∈I, whereP ∗ (H)denotes the family of all nonempty subsets ofH. A subsetK⊆His called asubhyperstructureofHif for everyi∈Iand every(x 1 , . . . , x n i )∈ K n i one has ? i (x 1 , . . . , x n i )⊆K. LetAbe a nonempty parameter set. A (non-null) soft set overHis a pair(F, A)with F:A→P ∗ (H),Supp(F, A) :=a∈A:F(a)6 =∅6=∅. Then(F, A)is called aSoft HyperStructure overHif for everya∈Supp(F, A), the subset F(a)⊆His a subhyperstructure ofH. Definition 4.16.2(Soft SuperHyperStructure).LetSH=SH(H,F)be a superhyperstructure with alevelled universeH 〈m〉 m≥0 and a family of superhyperoperations F=F j j∈J ,F j :H 〈` j,1 〉 ×·×H 〈` j,n j 〉 −→P ∗ ( H 〈r j 〉 ) , where` j,1 , . . . , ` j,n j , r j ≥0. LetAbe a nonempty parameter set. Asoft set over the levelled universeis a mapping S:A−→ ∏ m≥0 P ( H 〈m〉 ) ,a7−→ ( S 〈m〉 a ) m≥0 , and its support is Supp(S) :=a∈A:∃m≥0, S 〈m〉 a 6 =∅. For eachj∈J, extendF j to subsets by the usual set-extension: forX k ⊆H 〈` j,k 〉 , F j (X 1 , . . . , X n j ) := ⋃ (x 1 ,...,x n j )∈X 1 ×·×X n j F j (x 1 , . . . , x n j ). The soft setSis called aSoft SuperHyperStructure overSH(H,F)if for everya∈Supp(S)and everyj∈Jone has the levelwise closure condition F j ( S 〈` j,1 〉 a , . . . , S 〈` j,n j 〉 a ) ⊆S 〈r j 〉 a . Equivalently, for eacha∈Supp(S), the family ( S 〈m〉 a ) m≥0 determines a sub-superhyperstructure ofSH(H,F)under the restrictions of all operationsF j . 98 Chapter 4. Applications of Soft Set Example 4.16.3(A soft superhyperstructure for a multi-level academic program catalogue). Let the base (level-0) universeH 〈0〉 be the set of courses H 〈0〉 =c 1 , c 2 , c 3 , c 4 , interpreted as (say)Linear Algebra(c 1 ),Discrete Math(c 2 ),Machine Learning(c 3 ), andDatabases (c 4 ). Define level-1objects ascourse bundles(nonempty sets of courses): H 〈1〉 :=P ∗ ( H 〈0〉 ) , and level-2objects asprogram tracks(nonempty sets of bundles): H 〈2〉 :=P ∗ ( H 〈1〉 ) . Consider two superhyperoperations (soJ=1,2): F 1 :H 〈0〉 ×H 〈0〉 −→P ∗ ( H 〈1〉 ) ,F 1 (x, y) := x, y , which forms a 2-course bundle, and F 2 :H 〈1〉 ×H 〈1〉 −→P ∗ ( H 〈2〉 ) ,F 2 (B 1 , B 2 ) := B 1 , B 2 , which forms a track consisting of two bundles. LetSH=SH(H,F)withF=F 1 , F 2 . Let the parameter set be A=AI,DB, representing two departments that curate their own multi-level catalogues. Define a soft set over the levelled universe, S:A−→P(H 〈0〉 )×P(H 〈1〉 )×P(H 〈2〉 ),a7−→ ( S 〈0〉 a , S 〈1〉 a , S 〈2〉 a ) , by S 〈0〉 AI =c 1 , c 3 , S 〈1〉 AI = c 1 , c 3 , S 〈2〉 AI = c 1 , c 3 , S 〈0〉 DB =c 2 , c 4 , S 〈1〉 DB = c 2 , c 4 , S 〈2〉 DB = c 2 , c 4 . Verification of the closure conditions.Fora=AIwe have F 1 ( S 〈0〉 AI , S 〈0〉 AI ) = ⋃ x,y∈c 1 ,c 3 x, y ⊆ c 1 , c 3 =S 〈1〉 AI , and F 2 ( S 〈1〉 AI , S 〈1〉 AI ) = c 1 , c 3 =S 〈2〉 AI . The same argument holds fora=DB. Hence, for eacha∈Aand each operationF j , F j ( S 〈` j,1 〉 a , . . . , S 〈` j,n j 〉 a ) ⊆S 〈r j 〉 a , soSis a Soft SuperHyperStructure overSH(H,F). Real-life interpretation.Each department parametera∈Aselects its admissible courses (level0), the bundles it permits (level1), and the tracks it offers (level2), while ensuring that combining allowed items via the curriculum-construction rulesF 1 andF 2 stays within the department’s own catalogue. 99 Chapter 4. Applications of Soft Set 4.17 Soft Graph Neural Networks AGraph Neural Network (GNN)learns node- or graph-level representations by iterative message passing, aggregating neighbors’ features to predict labels or properties [296–301]. ASoft Graph Neural Network (SGNN)is a parameter-indexed family of GNN-based selections, where each context induces a soft set of chosen nodes via thresholds. LetG= (V, E)be a finite graph (directed or undirected) with a node-feature mapX:V→R d . In many learning tasks, a graph neural network (GNN) produces, for each nodev∈V, ascores(v)∈[0,1](e.g., a class- probability after a sigmoid/softmax). For our purposes, it suffices to treat a GNN as a black box that, given(G, X)and a choice of model/context parameters, returns such a score function. Definition 4.17.1(Soft Graph Neural Network (SGNN)).LetG= (V, E)be a finite graph with node featuresX:V→R d . LetAbe a nonempty set ofsoft parameters(contexts), such as: task modes, expert profiles, time indices, hyperparameter regimes, or view-definitions. ASoft Graph Neural Network(SGNN) on(G, X)with parameter setAis a family N= (s a , τ a ) a∈A , where for eacha∈A: •s a :V→[0,1]is a node-scoring function produced by a (possibly shared-weight) GNN under contexta(for instances a =Readout◦Encoder a (G, X)); •τ a ∈[0,1]is a (possiblya-dependent) decision threshold. Definition 4.17.2(Soft set induced by an SGNN).LetN=(s a , τ a ) a∈A be an SGNN as in Definition 4.17.1. Define a mapping F N :A−→P(V)byF N (a) :=v∈V|s a (v)≥τ a . Then(F N , A)is called theSGNN-induced soft set(of selected nodes). Theorem 4.17.3(Soft-set structure and well-definedness of SGNN outputs).LetN=(s a , τ a ) a∈A be an SGNN on(G, X). Then the mappingF N :A→P(V)in Definition 4.17.2 is well-defined, and hence(F N , A)is a (crisp) soft set over the universeV. Proof.Fixa∈A. By definition,s a :V→[0,1]andτ a ∈[0,1]. Therefore the predicate “s a (v)≥τ a ” is meaningful for everyv∈V, and the set F N (a) =v∈V|s a (v)≥τ a is a subset ofV. Since this holds for everya∈A, the rulea7→F N (a)defines a single-valued mappingF N :A→P(V). Consequently,(F N , A)is a soft set over the universeV. Theorem 4.17.4(Representation of any soft set by a trivial SGNN).Let(F, A)be any (crisp) soft set over a finite universeV. Then there exists an SGNNN=(s a , τ a ) a∈A on(G, X)(for any fixed graphGon vertex setVand any featuresX) such thatF N (a) =F(a)for alla∈A. 100 Chapter 4. Applications of Soft Set Proof.Define, for eacha∈A, the score functions a :V→[0,1]by the indicator rule s a (v) := 1, v∈F(a), 0, v/∈F(a), and setτ a := 1 2 . ThenF N (a) =v∈V|s a (v)≥τ a =F(a). Such(s a , τ a )can be realized by a degenerate “network” that ignores(G, X)and outputs constants. Hence an SGNN can reproduce any given soft set, so the SGNN formalism generalizes soft sets. Remark 4.17.5(Edge-version).One may analogously define an SGNN-induced soft set of edgesby lettings E a :E→[0,1]be an edge-score function (e.g., attention weights) and defining F E N (a) =ε∈E|s E a (ε)≥τ E a , yielding a soft set over universeE. 4.18 HyperSoft Graph Neural Network HyperSoft Graph Neural Network maps multi-attribute parameter tuples to GNN-based node selections, yielding hypersoft sets over graph vertices. Definition 4.18.1(Hypersoft parameter space).Letm≥1and letA 1 , . . . , A m be nonempty attribute domains(parameter groups), e.g., A 1 =task modes, A 2 =expert profiles, A 3 =time contexts, A 4 =hyperparameter regimes,etc. Define thehypersoft parameter spaceby the Cartesian product C:=A 1 ×A 2 ×·×A m . An elementa∈Cis anm-tuplea= (a 1 , . . . , a m )witha i ∈A i . Definition 4.18.2(HyperSoft Graph Neural Network (HSGNN)).LetG= (V, E)be a finite graph and letX:V→R d be a node-feature map. Fix a hypersoft parameter spaceC= ∏ m i=1 A i as in Definition 4.18.1. AHyperSoft Graph Neural Network(HSGNN) on(G, X)with parameter spaceCis a family N= (s a , τ a ) a∈C , where, for eacha∈C: •s a :V→[0,1]is a node-scoring function produced by a (possibly shared-weight) GNN under contexta; concretely, one may fix an architecture Φ : (G, X, θ)7−→Φ(G, X;θ)∈[0,1] V and a parameter-selection mapθ:C →Θand sets a (·) := Φ(G, X;θ(a))(·); •τ a ∈[0,1]is a (possibly context-dependent) decision threshold. 101 Chapter 4. Applications of Soft Set Definition 4.18.3(Hypersoft set induced by an HSGNN).LetN=(s a , τ a ) a∈C be an HS- GNN as in Definition 4.18.2. Define F N :C −→P(V)byF N (a) :=v∈V|s a (v)≥τ a . Then(F N ,C)is called theHSGNN-induced hypersoft set(of selected nodes) over the universe V. Theorem 4.18.4(Hypersoft-set structure and well-definedness).LetNbe an HSGNN on (G, X)with hypersoft parameter spaceC. Then the induced mappingF N in Definition 4.18.3 is well-defined and determines a hypersoft set over the universeV, i.e., (F N ,C)is a hypersoft set onV. Proof.Fixa∈ C. By Definition 4.18.2,s a :V→[0,1]is a function andτ a ∈[0,1]is a scalar. Hence the subset F N (a) =v∈V|s a (v)≥τ a is uniquely determined and satisfiesF N (a)⊆V. Therefore the assignmenta7→F N (a)defines a functionF N :C → P(V), so(F N ,C)is a hypersoft set by the definition of hypersoft sets (mapping from the Cartesian product parameter space into a powerset). This also proves well- definedness. 4.19 Soft Natural Languages Anatural languageis a human communication system with grammar and meaning, enabling com- positional expression and context-dependent interpretation [302–305]. A soft natural language models ambiguous linguistic phenomena by mapping contexts/parameters to sets of admissible interpretations, enabling structured uncertainty-aware processing. LetΣbe a (finite or count- able) alphabet and letΣ ∗ denote the set of all finite strings overΣ. Fix a nonempty set U⊆Σ ∗ , whose elements are regarded as linguistic objects (e.g., utterances, sentences, queries, or docu- ments). LetEbe a nonempty set oflinguistic parameters(contexts), whose elements may encode a natural language label (English, Japanese, etc.), a dialect, a register (formal/informal), a domain (medical/legal), or a time slice. Anacceptability relationis any binary relation |=⊆U×E, whereu|=eis read as “the linguistic objectuis acceptable under parametere”. 102 Chapter 4. Applications of Soft Set Definition 4.19.1(Soft Natural Languages).LetUbe a linguistic universe and letEbe a parameter set as above. Fix a nonempty subsetA⊆Eof parameters and an acceptability relation|=⊆U×A. Define the mapping F |= :A−→P(U),F |= (a) :=u∈U|u|=a. The pair SNL:= (F |= , A) is called aSoft Natural Languagesstructure onU(with parameter setA). For eacha∈A, the setF |= (a)is the collection of utterances judged acceptable under the linguistic contexta. Theorem 4.19.2(Soft-set structure and well-definedness).Every Soft Natural Languages struc- tureSNL= (F |= , A)in Definition 4.19.1 is a (crisp) soft set over the universeU. Equivalently, F |= is a well-defined mappingA→P(U). Proof.Fix anya∈A. By construction, F |= (a) =u∈U|u|=a is a subset ofU, henceF |= (a)∈ P(U). Since this holds for everya∈A, the rulea7→F |= (a) defines a single-valued mappingF |= :A→P(U). Therefore(F |= , A)is a soft set overUin the sense of Molodtsov. Theorem 4.19.3(Representation of soft sets as Soft Natural Languages).Let(F, A)be any soft set over the linguistic universeU(i.e.,F:A→P(U)). Define a relation|= F ⊆U×Aby u|= F a:⇐⇒u∈F(a). Then the induced mappingF |= F satisfiesF |= F =F, hence(F, A)is exactly the Soft Natural Languages structure generated by|= F . Proof.For eacha∈A, F |= F (a) =u∈U|u|= F a=u∈U|u∈F(a)=F(a). ThusF |= F =Fpointwise onA, proving the claim. 4.20 Softn-SuperHyperGraphs Ann-SuperHyperGraph is a hierarchical hypergraph in which vertices are built by iterating the nonempty powerset constructionntimes on a finite base set, and each superhyperedge is a nonempty family of suchn-supervertices encoding higher-order interactions [15, 306, 307]. A softn-SuperHyperGraph is a parameterized collection of induced sub-n-SuperHyperGraphs, where each parameter selects a subset ofn-supervertices together with a compatible subset of superhyperedges whose endpoints lie entirely within the selected vertices [308,309]. 103 Chapter 4. Applications of Soft Set Definition 4.20.1(Iterated powerset and iterated nonempty powerset).[292,310,311] LetH be a nonempty set. Define thek-th iterated powersetP k (H)recursively by P 0 (H) :=H,P k+1 (H) :=P ( P k (H) ) (k≥0), whereP(·)denotes the usual powerset. Define thek-th iterated nonempty powersetP k ∗ (H)by omitting∅at every level: P 0 ∗ (H) :=H,P k+1 ∗ (H) :=P ( P k ∗ (H) ) \∅(k≥0). Definition 4.20.2(n-SuperHyperGraph).[15,307,312,313] LetV 0 be a finite nonemptybase vertex set, and letn≥1be an integer. Set V n (V 0 ) :=P n−1 ∗ (V 0 ) (⊆P n−1 (V 0 )). Elements ofV n (V 0 )are calledn-supervertices. Ann-SuperHyperGraphis a pair SHG (n) = (V, E), where V⊆V n (V 0 )is finite and nonempty,E⊆P(V)\∅. Each element ofEis called ann-superhyperedge. (Thus every superhyperedge is a nonempty subset of then-supervertex setV.) In particular, forn= 1one hasV 1 (V 0 ) =V 0 , so SHG (1) is an ordinary (finite) hypergraph onV 0 . Forn= 2, vertices are nonempty subsets ofV 0 (groups of base vertices), and edges are families of such groups. Definition 4.20.3(Softn-SuperHyperGraph).[314, 315] Let SHG (n) = (V, E)be ann- SuperHyperGraph and letCbe a nonempty set of parameters. Asoftn-SuperHyperGraph (over SHG (n) with parameter setC) is a5-tuple ( V, E, C, A, B ) , where A:C−→P(V),B:C−→P(E), such that for eachc∈Cthe pair ( A(c), B(c) ) is a (parameter-induced) sub-superhypergraph of SHG (n) , i.e., A(c)⊆V,B(c)⊆e∈E:e⊆A(c). Example 4.20.4(A soft2-superhypergraph for university course grouping).Let the base vertex set be a small set of students V 0 =s 1 , s 2 , s 3 , s 4 , s 5 . 104 Chapter 4. Applications of Soft Set SinceV 2 (V 0 ) =P ∗ (V 0 ), a2-supervertex is a nonempty group of students. Define a2-SuperHyperGraph SHG (2) = (V, E)by V= v 1 =s 1 , s 2 , v 2 =s 2 , s 3 , v 3 =s 4 , v 4 =s 5 , v 5 =s 3 , s 4 ⊆V 2 (V 0 ), and by specifying superhyperedges as families of these student-groups: E= e 1 =v 1 , v 2 , v 5 , e 2 =v 2 , v 3 , e 3 =v 3 , v 4 , e 4 =v 1 , v 4 ⊆P(V)\∅. Interpretation: eachv i is a base study-group, and eache∈Eis a higher-order collaboration constraint among several groups (e.g., a shared project, lab session, or timetable coupling). LetC=AI,DB,Mathbe course-topic parameters. Define A:C→P(V),B:C→P(E), by selecting, for each topic, the relevant student-groups and the collaboration constraints among them: A(AI) =v 1 , v 2 , v 5 ,B(AI) =e 1 , A(DB) =v 2 , v 3 ,B(DB) =e 2 , A(Math) =v 3 , v 4 ,B(Math) =e 3 . Then for everyc∈Cwe haveA(c)⊆Vand B(c)⊆e∈E:e⊆A(c), so(A(c), B(c))is a sub-superhypergraph of SHG (2) . Hence ( V, E, C, A, B ) is a soft2-superhypergraph, encoding topic-dependent selections of student-groups and their higher-order collaboration constraints. 4.21 Recursive Soft SuperHyperGraph An(n, k)-recursive SuperHyperGraph has level-nsupervertices (iterated powersets) and depth-k recursive edges that may include supervertices and lower-level edges as elements. We restrict to well-founded recursive superhyperedges (no membership cycles). Definition 4.21.1((n, k)-recursive SuperHyperGraph).[316] Fix a base (ground) setV 0 and letn, k∈N∪0. (Iterated powersets).Define the iterated powersets by P 0 (V 0 ) =V 0 ,P n+1 (V 0 ) =P ( P n (V 0 ) ) (n≥0). A(n, k)-recursive SuperHyperGraphis a pair RSHG (n,k) = (V, E) satisfying: 105 Chapter 4. Applications of Soft Set (i)(Hierarchical supervertex set).V⊆P n (V 0 ). (i)(Recursive superhyperedge family).E⊆2 V,k \∅, where2 V,k is the depth-kpowerset universe constructed fromS=Vas in Definition??. To express the requirement that a recursive superhyperedge uses only vertices from a chosen subset, we define avertex-supportoperator supp k :P 〈k〉 (V)−→P(V) by recursion: supp 0 (v) :=v(v∈V),supp k+1 (X) := ⋃ Y∈X supp k (Y) ( X∈P 〈k+1〉 (V) ) . Thus, supp k (x)is the set of all base vertices that appear anywhere inside the nested objectx. Remark 4.21.2.IfW⊆V, then for anyx∈P 〈k〉 (V)we have x∈P 〈k〉 (W)=⇒supp k (x)⊆W. Conversely, if onedefinesthe restriction of recursive edges toWby E W :=e∈E:supp k (e)⊆W, thenE W consists precisely of those recursive edges ofEthat only use vertices fromW. We recall that an(n, k)-recursive SuperHyperGraph is a pair RSHG (n,k) = (V, E)whereV⊆ P n (V 0 )for a fixed ground setV 0 andE⊆P 〈k〉 (V)\∅. Definition 4.21.3((n, k)-recursive soft superhypergraph).Let RSHG (n,k) = (V, E)be an (n, k)-recursive SuperHyperGraph, and letCbe a nonempty set of parameters. An(n, k)- recursive soft SuperHyperGraph(over RSHG (n,k) with parameter setC) is a quintuple S:= (V, E, C, A, B), where A:C→P(V),B:C→P(E), such that for everyc∈Cthe pair ( A(c), B(c) ) is a sub-(n, k)-recursive SuperHyperGraph of(V, E)in the following sense: (RS1) (Vertex inclusion)A(c)⊆V; (RS2) (Recursive-edge compatibility) for everye∈B(c)one has supp k (e)⊆A(c). Equivalently, B(c)⊆E A(c) :=e∈E:supp k (e)⊆A(c). 106 Chapter 4. Applications of Soft Set Remark 4.21.4.Whenk= 1, each recursive edgee∈ P 〈1〉 (V) =P(V)is simply a subset ofV, and supp 1 (e) = ⋃ v∈e v=e. Hence (RS2) becomes the usual induced-edge condition e⊆A(c). Theorem 4.21.5(Simultaneous generalization).The concept in Definition 4.21.3 generalizes both (a)(n, k)-recursive SuperHyperGraphs, and (b)softn-SuperHyperGraphs (soft SuperHyperGraphs in the non-recursive sense). Proof.(a) Let RSHG (n,k) = (V, E)be any(n, k)-recursive SuperHyperGraph. Take the singleton parameter setC:=∗and define A(∗) :=V,B(∗) :=E. Then (RS1) holds trivially. For (RS2), everye∈B(∗) =Esatisfies supp k (e)⊆V=A(∗)by definition of supp k . Hence(V, E, C, A, B)is an(n, k)-recursive soft SuperHyperGraph. More- over, forgetting the (trivial) parameterization recovers exactly(V, E). (b) Let(V, E, C, A, B)be an(n,1)-recursive soft SuperHyperGraph. Sincek= 1, we have E⊆ P 〈1〉 (V)\ ∅=P(V)\ ∅, so each edgee∈Eis a nonempty subset ofV. By Remark 4.21.4, condition (RS2) becomes e∈B(c)=⇒e⊆A(c), equivalently, B(c)⊆e∈E:e⊆A(c). Thus(V, E, C, A, B)is precisely a softn-SuperHyperGraph in the standard (non-recursive) sense. Conversely, any softn-SuperHyperGraph(V, E, C, A, B)(withE⊆ P(V)\∅andB(c)⊆ e∈E:e⊆A(c)) is an(n,1)-recursive soft SuperHyperGraph because supp 1 (e) =efor all e⊆V. 4.22 Hierarchical Soft SuperHyperGraph A hierarchical superhypergraph permits vertices from several powerset levels and allows edges to joinmixed-levelvertices, while enforcing a downward-closure coherence principle [208]. Definition 4.22.1(Nonempty powerset tower and hierarchical universe).LetV 0 be a finite nonempty base set and fixr∈N 0 . Define anonempty powerset tower(P 〈k〉 (V 0 )) r k=0 by P 〈0〉 (V 0 ) :=V 0 ,P 〈k+1〉 (V 0 ) :=P ( P 〈k〉 (V 0 ) ) \∅(0≤k < r). Set thehierarchical universe U r (V 0 ) := r ⋃ k=0 P 〈k〉 (V 0 ). Forx∈U r (V 0 ), define itslevelby `(x) :=mink∈0,1, . . . , r:x∈P 〈k〉 (V 0 ). 107 Chapter 4. Applications of Soft Set Definition 4.22.2(Downward closure).LetW⊆U r (V 0 ). Define recursively D 0 (W) :=W,D t+1 (W) :=D t (W)∪ ⋃ X|X∈D t (W), `(X)≥1(t≥0). Since the tower height isr, the sequence stabilizes by stepr; we define thedownward closureof Wby dcl(W) :=D r (W). Then dcl(W)is the smallest subset ofU r (V 0 )that containsWand satisfies: ifX∈dcl(W)with `(X)≥1, thenX⊆dcl(W). Definition 4.22.3(Hierarchical SuperHyperGraph of heightr).LetV 0 be a finite nonempty base set and fixr∈N 0 . LetU r (V 0 )and`(·)be as in Definition 4.22.1. Ahierarchical SuperHy- perGraph of heightronV 0 is a pair H 〈r〉 = (V, E) satisfying: (H1)(Hierarchical vertex set).Vis a finite nonempty set withV⊆U r (V 0 ). (H2)(Cross-level hyperedges).E⊆P(V)\∅. (H3)(Coherence / downward closure).IfX∈Vand`(X)≥1, thenX⊆V. For eachk∈0, . . . , r, define thek-th layer V k :=x∈V:`(x) =k,so thatV= ̇ ⋃ r k=0 V k . Definition 4.22.4(Hierarchical Soft SuperHyperGraph).LetH 〈r〉 = (V, E)be a hierarchical SuperHyperGraph of heightron a base setV 0 in the sense of Definition 4.22.3. LetCbe a nonempty set of parameters. Ahierarchical soft SuperHyperGraph(of heightr) overH 〈r〉 is a quintuple S= ( V, E, C, A, B ) , where A:C−→P(V),B:C−→P(E), such that for everyc∈C: (HS1)(Downward-closed slice vertices).IfX∈A(c)and`(X)≥1, thenX⊆A(c). (HS2)(Induced slice edges).B(c)⊆e∈E:e⊆A(c). For eachc∈C, the pair S[c] := ( A(c), B(c) ) is called theparameter slice(a sub-hierarchical-superhypergraph) atc. 108 Chapter 4. Applications of Soft Set Proposition 4.22.5(Each slice is a hierarchical SuperHyperGraph).LetS= (V, E, C, A, B) be a hierarchical soft SuperHyperGraph as in Definition 4.22.4. Then for everyc∈C, the slice S[c] = (A(c), B(c))is a hierarchical SuperHyperGraph (possibly empty-vertex if one allows it; if one requires nonempty vertex sets, assumeA(c)6=∅). Proof.Fixc∈C. SinceA(c)⊆V⊆ U r (V 0 ), the vertex condition holds. AlsoB(c)⊆E⊆ P(V)\∅and by (HS2) eache∈B(c)satisfiese⊆A(c), henceB(c)⊆P(A(c))\∅. Finally, (HS1) is exactly the coherence (downward-closure) axiom for the slice. Theorem 4.22.6(Hierarchical SuperHyperGraphs are special cases).LetH 〈r〉 = (V, E)be a hierarchical SuperHyperGraph. Define a singleton parameter setC:=c 0 and maps A(c 0 ) :=V,B(c 0 ) :=E. ThenS= (V, E, C, A, B)is a hierarchical soft SuperHyperGraph, and its unique sliceS[c 0 ] coincides withH 〈r〉 . Proof.SinceA(c 0 ) =V, the slice-vertex downward-closure (HS1) holds becauseVsatisfies (H3). AlsoB(c 0 ) =E⊆ e∈E:e⊆V, so (HS2) holds. ThusSis hierarchical soft, and S[c 0 ] = (V, E) =H 〈r〉 . Theorem 4.22.7(Softn-SuperHyperGraphs embed into hierarchical soft SuperHyperGraphs). Fixn≥1and letV 0 be a finite base set. LetSHG (n) = (V n , E)be ann-SuperHyperGraph whose vertex set satisfiesV n ⊆P 〈n〉 (V 0 ), and let H 〈n〉 := ( dcl(V n ), E ) be the hierarchical structure obtained by closing the vertex set downward (Definition 4.22.2), keeping the same hyperedge familyE⊆P(V n )\∅. Let(V n , E, C, A n , B)be a softn-SuperHyperGraph (i.e.,A n :C→ P(V n )andB:C→ P(E) withB(c)⊆e∈E:e⊆A n (c)). Define A(c) :=dcl ( A n (c) ) ⊆dcl(V n ),B(c) :=B(c)⊆E(c∈C). Then S= ( dcl(V n ), E, C, A, B ) is a hierarchical soft SuperHyperGraph of heightnoverH 〈n〉 . Moreover, restricting each slice to the top layer recovers the original softn-SuperHyperGraph: A n (c) =A(c)∩V n ,B(c) =B(c) (c∈C). Proof.First, dcl(V n )⊆U n (V 0 )by construction, and it is downward closed by Definition 4.22.2. HenceH 〈n〉 = (dcl(V n ), E)satisfies (H1) and (H3). SinceE⊆ P(V n )\∅andV n ⊆dcl(V n ), we also haveE⊆P(dcl(V n ))\∅, so (H2) holds. Now fixc∈C. By definition,A(c) =dcl(A n (c))is downward closed, so (HS1) holds. For (HS2), take anye∈B(c). In the original softn-SuperHyperGraph,e⊆A n (c). SinceA n (c)⊆A(c), it follows thate⊆A(c), hencee∈ e ′ ∈E:e ′ ⊆A(c). ThereforeB(c)⊆ e∈E:e⊆A(c), establishing (HS2). ThusSis a hierarchical soft SuperHyperGraph. Finally, sinceA n (c)⊆V n andV n is precisely the top-level part of dcl(V n ), downward closure adds only lower-level constituents; henceA(c)∩V n =A n (c). The edge component is unchanged. 109 Chapter 5 Soft Decision-Making In this chapter, we examine several soft decision-making methods. 5.1 Soft decision-making Soft decision-making ranks finitely many alternatives by weighted counts of satisfied parameters in a soft set, and selects the argmax scorers. Related frameworks includefuzzy decision-making [317,318] andneutrosophic decision-making[319,320]. Definition 5.1.1(Soft decision-making model induced by a soft set).LetUbe a nonempty finiteset of alternatives (objects) and letEbe a nonempty set of parameters (criteria). Fix a nonempty subsetA⊆Eand a soft set(F, A)overU, i.e., F:A−→P(U). Letw:A→[0,∞)be a weight (importance) function. For each alternativeu∈Udefine the(crisp) satisfaction indicatorundera∈Aby χ F (u, a) := 1, u∈F(a), 0, u/∈F(a). Thesoft decision scoreofu(with respect to(F, A)andw) is the real number S (F,A),w (u) := ∑ a∈A w(a)χ F (u, a). The inducedsoft decision correspondenceis DM(F, A;w) :=arg max u∈U S (F,A),w (u)⊆U, i.e., the set of all alternatives having maximum score. Equivalently, one may define the induced (pre)order (F,A),w onUby u (F,A),w v⇐⇒S (F,A),w (u)≥S (F,A),w (v), and thenDM(F, A;w)is the set of (F,A),w -maximal elements. 111 Chapter 5. Soft Decision-Making Theorem 5.1.2(Soft-set structure and well-definedness).In the setting of Definition 5.1.1: (i)Soft-set structure.The data used by the decision model is exactly a soft set(F, A)overU (together with a weight mapw). (i)Well-defined score.The mappingS (F,A),w :U→Ris well-defined (unique value for each u∈U). (i)Existence of optimal decisions.The decision correspondenceDM(F, A;w)is well-defined and nonempty. (iv)Invariance under extension to the full parameter set.Define the extension ̃ F:E→ P(U)by ̃ F(e) = F(e), e∈A, ∅,e∈E . Then the decision outcome computed from(F, A)equals the outcome computed from ̃ Fre- stricted toA; in particular, adding “unused” parameters with empty images does not change DM(F, A;w). Proof.(i) By assumption,A⊆Eis nonempty andF:A→P(U); hence(F, A)is, by definition, a soft set overU. The additional mapw:A→[0,∞)only provides importance weights and does not alter the underlying soft-set structure. (i) Fixu∈U. For eacha∈A, the statementu∈F(a)is unambiguous, soχ F (u, a)∈ 0,1 is uniquely determined. SinceAis finite andw(a)χ F (u, a)∈[0,∞)for eacha, the finite sum S (F,A),w (u) = ∑ a∈A w(a)χ F (u, a)is a uniquely determined real number. ThusS (F,A),w :U→R is well-defined. (i) BecauseUis finite andS (F,A),w (u)∈Ris well-defined for everyu∈Uby (i), the max- imum value max u∈U S (F,A),w (u)exists. Therefore the argmax setDM(F, A;w) =u∈U: S (F,A),w (u) =max v∈U S (F,A),w (v)is a well-defined subset ofUand is nonempty. (iv) Fora∈Awe have ̃ F(a) =F(a)by definition of ̃ F, henceχ ̃ F (u, a) =χ F (u, a)for allu∈U anda∈A. Therefore, for everyu∈U, S ( ̃ F| A ,A),w (u) = ∑ a∈A w(a)χ ̃ F (u, a) = ∑ a∈A w(a)χ F (u, a) =S (F,A),w (u). Thus the score functions coincide onU, and consequently their argmax sets coincide:DM( ̃ F| A , A;w) = DM(F, A;w). 112 Chapter 5. Soft Decision-Making 5.2 HyperSoft TOPSIS and SuperHyperSoft TOPSIS TOPSIS ranks alternatives by distance to positive ideal and negative ideal points, using normal- ized weighted criteria values [29,321–323]. HyperSoft TOPSIS ranks alternatives using TOPSIS on multi-attribute value tuples (hypersoft parameters) with weighted distances to ideal solu- tions. SuperHyperSoft TOPSIS extends HyperSoft TOPSIS by allowing each attribute to be a value-subset, enabling set-valued criteria tuples. Definition 5.2.1(Hypersoft criterion domain).LetU=u 1 , . . . , u n be a finite set of al- ternatives. LetA 1 , . . . ,A k be pairwise-disjoint attribute-value sets (domains), and define the hypersoft parameter space C:=A 1 ×A 2 ×·×A k . A finite set ofcriteria-tuplesis a subset Λ =λ 1 , . . . , λ m ⊆C, where eachλ j = (a 1j , . . . , a kj )specifies a combined multi-attribute value tuple. Definition 5.2.2(Numeric hypersoft evaluation).Anumeric hypersoft evaluationis a map x:U×Λ→R ≥0 ,(u i , λ j )7→x ij . Equivalently,xis thehypersoft decision matrixX= (x ij )∈R n×m ≥0 . (Crisp hypersoft set as a special case.) If one is given a crisp hypersoft setG: Λ→P(U), then it induces a numeric evaluation via x ij :=1 u i ∈G(λ j ) ∈0,1. More generally, if evaluations are given in a graded/uncertain hypersoft form (fuzzy, neutro- sophic, picture fuzzy, interval-valued, etc.), one first applies ascoremap Score:(grade object)→ R ≥0 and setsx ij :=Score(grade ofu i atλ j ). Definition 5.2.3(HyperSoft TOPSIS).Assume the hypersoft decision matrixX= (x ij )∈ R n×m ≥0 is given. Letw= (w 1 , . . . , w m )be criterion weights with w j >0, m ∑ j=1 w j = 1. LetB⊆1, . . . , mbe the index set ofbenefitcriteria andC⊆1, . . . , mbe the index set of costcriteria, withB∪C=1, . . . , mandB∩C=∅. (1) Vector normalization.For eachj∈1, . . . , mdefine d j := √ √ √ √ n ∑ i=1 x 2 ij . 113 Chapter 5. Soft Decision-Making The normalized matrixR= (r ij )is r ij := x ij d j , d j >0, 0,d j = 0. (2) Weighted normalized matrix.DefineV= (v ij )by v ij :=w j r ij . (3) Hypersoft positive/negative ideal solutions.For each criterionj, define the ideal values v + j := max 1≤i≤n v ij , j∈B, min 1≤i≤n v ij , j∈C, v − j := min 1≤i≤n v ij , j∈B, max 1≤i≤n v ij , j∈C. LetA + := (v + 1 , . . . , v + m )andA − := (v − 1 , . . . , v − m ). (4) Separation measures.For each alternativeu i , define the (Euclidean) distances to the ideals: S + i := √ √ √ √ m ∑ j=1 (v ij −v + j ) 2 ,S − i := √ √ √ √ m ∑ j=1 (v ij −v − j ) 2 . (5) Closeness coefficient and ranking.Define thehypersoft TOPSIS closeness coefficient C i := S − i S + i +S − i , S + i +S − i >0, 1 2 ,S + i +S − i = 0, and rank alternatives by decreasingC i : u p <u q ⇐⇒C p ≥C q . Any total order refining<(e.g., by tie-breaking rules) is called aHyperSoft TOPSIS ranking. Definition 5.2.4(SuperHyperSoft parameter space and criteria family).LetU=u 1 , . . . , u n be a finite set of alternatives. LetA 1 , . . . , A k be pairwise-disjoint attribute-value sets (k≥1). Define thesuperhypersoft parameter space C SH :=P(A 1 )×P(A 2 )×·×P(A k ), whose elementsγ= (α 1 , . . . , α k )are tuples ofvalue-subsetsα t ⊆A t . A finite set ofsuper-criteria is a subset Γ =γ 1 , . . . , γ m ⊆C SH . 114 Chapter 5. Soft Decision-Making Definition 5.2.5(Numeric superhypersoft evaluation and decision matrix).Anumeric super- hypersoft evaluation(onΓ) is a map x SH :U×Γ→R ≥0 ,(u i , γ j )7→x SH ij . Equivalently,x SH is represented by thesuperhypersoft decision matrixX SH = (x SH ij )∈R n×m ≥0 . (Crisp/graded inputs.) If the raw information is given as a (crisp) SuperHyperSoft setF: Γ→ P(U), one may takex SH ij :=1 u i ∈F(γ j ) . If the raw information is graded (fuzzy, neutrosophic, interval-valued, etc.), assume a fixed score map Score toR ≥0 and setx SH ij :=Score(grade ofu i underγ j ). Definition 5.2.6(SuperHyperSoft TOPSIS).Assume the superhypersoft decision matrixX SH = (x SH ij )∈R n×m ≥0 is given. Letw= (w 1 , . . . , w m )be weights withw j >0and ∑ m j=1 w j = 1. Let B(benefit) andC(cost) be a partition of1, . . . , m. (1) Vector normalization.For eachjlet d SH j := √ √ √ √ n ∑ i=1 (x SH ij ) 2 ,r SH ij := x SH ij d SH j , d SH j >0, 0,d SH j = 0. LetR SH = (r SH ij ). (2) Weighted normalized matrix.DefineV SH = (v SH ij )by v SH ij :=w j r SH ij . (3) Ideal solutions.For eachjdefine v SH,+ j := max 1≤i≤n v SH ij , j∈B, min 1≤i≤n v SH ij , j∈C, v SH,− j := min 1≤i≤n v SH ij , j∈B, max 1≤i≤n v SH ij , j∈C. LetA SH,+ = (v SH,+ 1 , . . . , v SH,+ m )andA SH,− = (v SH,− 1 , . . . , v SH,− m ). (4) Separation measures.For each alternativeu i set S SH,+ i := √ √ √ √ m ∑ j=1 ( v SH ij −v SH,+ j ) 2 ,S SH,− i := √ √ √ √ m ∑ j=1 ( v SH ij −v SH,− j ) 2 . (5) Closeness coefficient and ranking.Define C SH i := S SH,− i S SH,+ i +S SH,− i , S SH,+ i +S SH,− i >0, 1 2 ,S SH,+ i +S SH,− i = 0, and rank alternatives by decreasingC SH i . 115 Chapter 5. Soft Decision-Making Definition 5.2.7(Singleton embedding of hypersoft parameters).LetC H :=A 1 ×·×A k be the hypersoft parameter space. Define the map ι:C H →C SH ,ι(a 1 , . . . , a k ) := (a 1 , . . . ,a k ). ForΛ =λ 1 , . . . , λ m ⊆C H , set Γ :=ι(Λ) =ι(λ 1 ), . . . , ι(λ m )⊆C SH . Theorem 5.2.8(SuperHyperSoft TOPSIS strictly generalizes HyperSoft TOPSIS).LetUbe a set of alternatives and letΛ =λ 1 , . . . , λ m ⊆A 1 ×·×A k be a hypersoft criterion family. Let x H :U×Λ→R ≥0 be a numeric hypersoft evaluation and defineΓ =ι(Λ)as in Theorem 5.2.7. Define a superhypersoft evaluationx SH :U×Γ→R ≥0 by x SH (u, ι(λ)) :=x H (u, λ),(u∈U, λ∈Λ). Fix the same weight vectorwand the same benefit/cost partition(B, C)for both methods. Then the SuperHyperSoft TOPSIS closeness coefficients equal the HyperSoft TOPSIS closeness coefficients: C SH i =C H i for alli∈1, . . . , n, and hence both methods produce the same ranking of alternatives. Proof.IndexΛ =λ 1 , . . . , λ m andΓ =γ 1 , . . . , γ m withγ j =ι(λ j ). By definition ofx SH , the two decision matrices coincide entrywise: x SH ij =x SH (u i , γ j ) =x SH (u i , ι(λ j )) =x H (u i , λ j ) =x H ij . Therefore, for each columnjthe normalization denominators are equal: d SH j = √ √ √ √ n ∑ i=1 (x SH ij ) 2 = √ √ √ √ n ∑ i=1 (x H ij ) 2 =d H j . Hencer SH ij =r H ij for alli, j, and thus also v SH ij =w j r SH ij =w j r H ij =v H ij . Since the weighted normalized matrices coincide, their componentwise maxima/minima overi coincide as well; thusA SH,+ =A H,+ andA SH,− =A H,− . Consequently, the separation measures are identical for eachi: S SH,± i = √ √ √ √ m ∑ j=1 (v SH ij −v SH,± j ) 2 = √ √ √ √ m ∑ j=1 (v H ij −v H,± j ) 2 =S H,± i . Plugging into the closeness coefficient formula yieldsC SH i =C H i for alli. Therefore the induced rankings coincide. 5.3 Soft, HyperSoft, and SuperHyperSoft AHP AHP is an MCDM method using pairwise comparisons to derive ratio-scale weights, aggregate priorities hierarchically, and rank alternatives [324–327]. Soft AHP applies pairwise comparison matrices to soft parameters, derives criteria weights and alternative priorities, then aggregates them into rankings. HyperSoft AHP extends Soft AHP by using multiple attribute value tuples as criteria, enabling richer parameterized pairwise evaluations for alternatives. SuperHyperSoft AHP generalizes HyperSoft AHP by allowing set valued attribute choices per criterion tuple, improving decision flexibility under uncertainty contexts. 116 Chapter 5. Soft Decision-Making Definition 5.3.1(Positive reciprocal (pairwise-comparison) matrix).Letn∈N. A matrix A= (a ij )∈R n×n is called apositive reciprocal matrixif a ij >0,a i = 1,a ij = 1 a ji (1≤i, j≤n). We denote byR n the set of all positive reciprocal matrices of sizen. Definition 5.3.2(Priority vector operator).LetA∈ R n . SinceAis a positive matrix, by the Perron–Frobenius theoremAhas a largest eigenvalueλ max (A)>0with a strictly positive eigenvector. Define thepriority vectorπ(A)∈R n >0 as any Perron eigenvector normalized to sum 1: A π(A) =λ max (A)π(A), n ∑ i=1 π i (A) = 1. (When the Perron eigenvalue is simple,π(A)is unique.) Remark 5.3.3(Consistency).A matrixA∈R n is (multiplicatively)consistentiffa ij a jk =a ik for alli, j, k. In that case there existsw∈R n >0 such thata ij =w i /w j , and thenπ(A)recovers wup to normalization. Definition 5.3.4(Soft AHP decision instance).LetU=u 1 , . . . , u n be a finite set of alterna- tives and letEbe a set of parameters (criteria). Fix asoft parameter setS=e 1 , . . . , e m ⊆E. ASoft AHP decision instanceis a tuple SAHP= (U, E, S, A (0) ,A (e) e∈S ), where (i)A (0) ∈R m is the criteria pairwise-comparison matrix indexed byS; (i) for eache∈S,A (e) ∈R n is the alternative pairwise-comparison matrix under criterione. Thecriteria weight vectorisw=π(A (0) )∈R m >0 , and thelocal alternative weightundere j is p (j) =π(A (e j ) )∈R n >0 . Theglobal priority(overall score) of alternatives is the vector P:= m ∑ j=1 w j p (j) ∈R n >0 ,soP i = m ∑ j=1 w j p (j) i . ASoft AHP rankingis any ordering ofUthat is nonincreasing inP i . 117 Chapter 5. Soft Decision-Making Definition 5.3.5(HyperSoft AHP decision instance).LetU=u 1 , . . . , u n be alternatives. LetA 1 , . . . ,A k be pairwise-disjoint attribute-value sets and define the hypersoft parameter space C H =A 1 ×·×A k . Fix a finitecriteria-tuples family Λ =λ 1 , . . . , λ m ⊆C H . AHyperSoft AHP decision instanceis a tuple HAHP= (U,A 1 , . . . ,A k ,Λ, A (0) ,A (λ) λ∈Λ ), whereA (0) ∈ R m compares the criteria-tuples inΛ, and for eachλ∈Λ,A (λ) ∈ R n compares alternatives under criterion-tupleλ. Definew=π(A (0) )∈R m >0 andp (j) =π(A (λ j ) )∈R n >0 . Theglobal priorityis P:= m ∑ j=1 w j p (j) ∈R n >0 , and alternatives are ranked by nonincreasingP i . Theorem 5.3.6(HyperSoft AHP generalizes Soft AHP).Every Soft AHP decision instance can be realized as a HyperSoft AHP decision instance, and under this realization both methods produce identical global priorities and rankings. Proof.LetSAHP= (U, E, S, A (0) ,A (e) e∈S )withS=e 1 , . . . , e m . Setk= 1and letA 1 := E. ThenC H =A 1 =E. DefineΛ :=S⊆Eand identifyλ j ≡e j . Now define a HyperSoft AHP instance by taking the same criteria matrixA (0) ∈ R m and, for eachλ=e∈Λ, setA (λ) :=A (e) . Then by construction, the criteria weight vectorw=π(A (0) ) is the same in both models, and each local vector satisfies π(A (λ j ) ) =π(A (e j ) ). Hence the global priority vectors coincide: P H = m ∑ j=1 w j π(A (λ j ) ) = m ∑ j=1 w j π(A (e j ) ) =P S . Therefore the induced rankings are identical. Definition 5.3.7(SuperHyperSoft AHP decision instance).LetU=u 1 , . . . , u n be alter- natives. LetA 1 , . . . , A k be pairwise-disjoint attribute-value sets and define the superhypersoft parameter space C SH =P(A 1 )×P(A 2 )×·×P(A k ). Fix a finitesuper-criteria family Γ =γ 1 , . . . , γ m ⊆C SH ,γ j = (α 1j , . . . , α kj ), α tj ⊆A t . 118 Chapter 5. Soft Decision-Making ASuperHyperSoft AHP decision instanceis SHAHP= (U, A 1 , . . . , A k ,Γ, A (0) ,A (γ) γ∈Γ ), whereA (0) ∈ R m compares the elements ofΓ, and for eachγ∈Γ,A (γ) ∈ R n compares alternatives under super-criterionγ. Letw=π(A (0) )∈R m >0 andp (j) =π(A (γ j ) )∈R n >0 . Theglobal priorityis P:= m ∑ j=1 w j p (j) ∈R n >0 , and alternatives are ranked by nonincreasingP i . Definition 5.3.8(Singleton embedding).LetC H =A 1 ×·×A k andC SH =P(A 1 )×·× P(A k ). Define ι:C H →C SH ,ι(a 1 , . . . , a k ) := (a 1 , . . . ,a k ). ForΛ =λ 1 , . . . , λ m ⊆C H , setΓ :=ι(Λ)and indexΓ =γ 1 , . . . , γ m byγ j =ι(λ j ). Theorem 5.3.9(SuperHyperSoft AHP generalizes HyperSoft AHP).Every HyperSoft AHP decision instance can be realized as a SuperHyperSoft AHP decision instance via the singleton embeddingι. Under this realization, both methods yield identical global priorities and rankings. Proof.LetHAHP= (U, A 1 , . . . , A k ,Λ, A (0) ,A (λ) λ∈Λ )withΛ =λ 1 , . . . , λ m . DefineΓ = ι(Λ)andγ j =ι(λ j )as in Theorem 5.3.8. Construct a SuperHyperSoft AHP instance by taking the same criteria matrixA (0) ∈R m and defining A (γ j ) :=A (λ j ) (j= 1, . . . , m). Then the criteria weight vector is the samew=π(A (0) )in both models, and each local alternative priority vector satisfies π(A (γ j ) ) =π(A (λ j ) ). Hence the global priority vectors coincide: P SH = m ∑ j=1 w j π(A (γ j ) ) = m ∑ j=1 w j π(A (λ j ) ) =P H . Therefore the induced rankings are identical. 5.4 Soft, HyperSoft, and SuperHyperSoft VIKOR VIKOR is an MCDM method ranking alternatives by compromise using group utility and in- dividual regret, controlled by parameter v [328–331]. Soft VIKOR ranks alternatives using soft parameters, computes best worst values, group utility S, individual regret R, compromise Q index. HyperSoft VIKOR replaces single parameters with attribute value tuples, then applies VIKOR normalization, S and R aggregation, Q ranking procedure. SuperHyperSoft VIKOR allows set valued attribute choices per criterion, embedding HyperSoft via singleton sets, pre- serving S R Q outputs exactly. 119 Chapter 5. Soft Decision-Making Notation 5.4.1(Alternatives, criteria-orientation, and weights).LetU=u 1 , . . . , u n be a finite set of alternatives and letm∈N. Acriterion(parameter) will always be equipped with an orientation τ∈ben,cost, meaning that larger values are preferred forbenand smaller values are preferred forcost. Aweight vectoron a finite criterion-familyC=c 1 , . . . , c m isw= (w 1 , . . . , w m )∈[0,1] m with ∑ m j=1 w j = 1. Definition 5.4.2(Ideal best/worst values).LetC=c 1 , . . . , c m be a criterion-family with orientationsτ j ∈ ben,cost. Letf:U×C→Rbe an evaluation function and writef ij := f(u i , c j ). For eachj∈1, . . . , mdefine theideal bestvaluef ∗ j andideal worstvaluef − j by (f ∗ j , f − j ) := ( max 1≤i≤n f ij ,min 1≤i≤n f ij ) , τ j =ben, ( min 1≤i≤n f ij ,max 1≤i≤n f ij ) , τ j =cost. Definition 5.4.3(Normalized loss (distance from the ideal)).Under the hypotheses of Theo- rem 5.4.2, define thenormalized lossd ij ∈[0,1]by d ij := 0,f ∗ j =f − j , f ∗ j −f ij f ∗ j −f − j , τ j =ben andf ∗ j 6=f − j , f ij −f ∗ j f − j −f ∗ j , τ j =cost andf ∗ j 6 =f − j . Equivalently,d ij = 0iffu i attains the ideal best on criterionc j (or the criterion is constant). Definition 5.4.4(Group utility and individual regret).Letd ij be as in Theorem 5.4.3 and let w∈[0,1] m with ∑ j w j = 1. Define for each alternativeu i : S i := m ∑ j=1 w j d ij andR i :=max 1≤j≤m (w j d ij ). Set S ∗ :=min 1≤i≤n S i , S − :=max 1≤i≤n S i ,R ∗ :=min 1≤i≤n R i , R − :=max 1≤i≤n R i . Definition 5.4.5(Compromise indexQ).Letv∈[0,1]be fixed (oftenv= 1/2). With S i , R i , S ∗ , S − , R ∗ , R − as in Theorem 5.4.4, define Q i :=v S i −S ∗ S − −S ∗ + (1−v) R i −R ∗ R − −R ∗ , using the convention that 0 0 := 0(so ifS − =S ∗ then the first fraction is set to0, and similarly forR − =R ∗ ). 120 Chapter 5. Soft Decision-Making Definition 5.4.6(Soft VIKOR decision instance and solution).LetEbe a parameter set and letS=e 1 , . . . , e m ⊆Ebe a finite soft-parameter set. ASoft VIKOR decision instanceis a tuple SVIKOR= (U, E, S, τ, w, f, v), where (i)τ:S→ben,costassigns an orientationτ j :=τ(e j )to each parameter; (i)w= (w 1 , . . . , w m )∈[0,1] m with ∑ m j=1 w j = 1is the criterion-weight vector; (i)f:U×S→Ris an evaluation function (decision matrix) withf ij :=f(u i , e j ); (iv)v∈[0,1]is the compromise coefficient. Computef ∗ j , f − j by Theorem 5.4.2, thend ij by Theorem 5.4.3, thenS i , R i by Theorem 5.4.4, and finallyQ i by Theorem 5.4.5. ASoft VIKOR rankingis any ordering ofUthat is nondecreasing inQ i . (Optionally) Letu (1) andu (2) be the first and second alternatives under theQ-ordering. Define DQ:= 1 n−1 . If (a)Q(u (2) )−Q(u (1) )≥DQ(acceptable advantage) and (b)u (1) is also best by Sor byR(acceptable stability), thenu (1) is called the(unique) compromise solution. Otherwise one may output a compromise set consisting of the top fewQ-alternatives. Definition 5.4.7(HyperSoft VIKOR decision instance).LetA 1 , . . . ,A k be (pairwise-disjoint) attribute-value sets and let C H :=A 1 ×·×A k be the hypersoft parameter domain. Fix a finite criteria-tuples familyΛ =λ 1 , . . . , λ m ⊆C H . AHyperSoft VIKOR decision instanceis a tuple HVIKOR= (U,A 1 , . . . ,A k ,Λ, τ, w, f, v), whereτ: Λ→ben,cost,w∈[0,1] m with ∑ j w j = 1,f:U×Λ→R, andv∈[0,1]. With indicesf ij :=f(u i , λ j )andτ j :=τ(λ j ), definef ∗ j , f − j ,d ij ,S i , R i , andQ i exactly as in Theorems 5.4.2 to 5.4.5. The ranking is nondecreasing inQ i . Theorem 5.4.8(HyperSoft VIKOR generalizes Soft VIKOR).Every Soft VIKOR decision instance can be realized as a HyperSoft VIKOR decision instance. Under this realization, all computed quantities(f ∗ j , f − j , d ij , S i , R i , Q i )coincide, hence the rankings (and compromise solu- tions/sets) coincide. 121 Chapter 5. Soft Decision-Making Proof.LetSVIKOR= (U, E, S, τ, w, f, v)withS=e 1 , . . . , e m . Setk= 1and defineA 1 :=E, soC H =A 1 =E. LetΛ :=S⊆Eand identifyλ j ≡e j . Define the HyperSoft instance by keeping the samevand the same weight vectorw, setting τ(λ j ) :=τ(e j ), and definingf(u, λ) :=f(u, e)under the identificationλ≡e. Then for everyi, jwe have identical entriesf ij in both models, hence the best/worst valuesf ∗ j , f − j from Theorem 5.4.2 coincide, and therefore the normalized lossesd ij from Theorem 5.4.3 coincide. With the samew, this forcesS i andR i from Theorem 5.4.4 to coincide, and consequentlyQ i from Theorem 5.4.5 coincides. Thus the induced rankings and any compromise outputs coincide. Definition 5.4.9(SuperHyperSoft VIKOR decision instance).LetA 1 , . . . , A k be (pairwise- disjoint) attribute-value sets and define the superhypersoft domain C SH :=P(A 1 )×P(A 2 )×·×P(A k ). Fix a finite familyΓ =γ 1 , . . . , γ m ⊆C SH ofset-valuedcriteria-tuples. ASuperHyperSoft VIKOR decision instanceis a tuple SHVIKOR= (U, A 1 , . . . , A k ,Γ, τ, w, f, v), whereτ: Γ→ben,cost,w∈[0,1] m with ∑ j w j = 1,f:U×Γ→R, andv∈[0,1]. Definef ∗ j , f − j ,d ij ,S i , R i , andQ i exactly as in Theorems 5.4.2 to 5.4.5 withγ j in place ofc j . Rank alternatives nondecreasingly byQ i . Definition 5.4.10(Singleton embedding).LetC H :=A 1 ×·×A k andC SH :=P(A 1 )×·× P(A k ). Define ι:C H →C SH ,ι(a 1 , . . . , a k ) := (a 1 , . . . ,a k ). ForΛ =λ 1 , . . . , λ m ⊆C H setΓ :=ι(Λ), indexed asγ j :=ι(λ j ). Theorem 5.4.11(SuperHyperSoft VIKOR generalizes HyperSoft VIKOR).Every HyperSoft VIKOR decision instance can be realized as a SuperHyperSoft VIKOR decision instance via the singleton embeddingι. Under this realization, all computed quantities(f ∗ j , f − j , d ij , S i , R i , Q i ) coincide, hence the rankings coincide. Proof.LetHVIKOR= (U, A 1 , . . . , A k ,Λ, τ, w, f, v)withΛ =λ 1 , . . . , λ m ⊆A 1 ×·×A k . LetΓ =ι(Λ)andγ j =ι(λ j )as in Theorem 5.4.10. Define the SuperHyperSoft instance by keeping the samevandw, setting τ(γ j ) :=τ(λ j ),f(u, γ j ) :=f(u, λ j ) (u∈U, j= 1, . . . , m). Then for everyi, j, the entriesf ij coincide under the identificationγ j ↔λ j . Thereforef ∗ j , f − j coincide (same extrema over the same numbers), henced ij coincide. With identical weightsw, the aggregatesS i andR i coincide, and thusQ i coincides. Consequently the induced rankings coincide. 122 Chapter 6 Conclusion In this book, we provided a survey-style overview of soft set theory and its major developments. We expect that the concepts reviewed here will stimulate further research, especially on algorithm design and applications in machine learning and related areas. 123 Disclaimer Funding This study did not receive any financial or external support from organizations or individuals. Acknowledgments We extend our sincere gratitude to everyone who provided insights, inspiration, and assistance throughout this research. We particularly thank our readers for their interest and acknowledge the authors of the cited works for laying the foundation that made our study possible. We also appreciate the support from individuals and institutions that provided the resources and infrastructure needed to produce and share this book. Finally, we are grateful to all those who supported us in various ways during this project. Data Availability This research is purely theoretical, involving no data collection or analysis. We encourage fu- ture researchers to pursue empirical investigations to further develop and validate the concepts introduced here. Ethical Approval As this research is entirely theoretical in nature and does not involve human participants or animal subjects, no ethical approval is required. Use of Generative AI and AI-Assisted Tools I use generative AI and AI-assisted tools for tasks such as English grammar checking, and I do not employ them in any way that violates ethical standards. 125 Chapter 6. Conclusion Conflicts of Interest The authors confirm that there are no conflicts of interest related to the research or its publica- tion. Disclaimer This work presents theoretical concepts that have not yet undergone practical testing or valida- tion. Future researchers are encouraged to apply and assess these ideas in empirical contexts. While every effort has been made to ensure accuracy and appropriate referencing, unintentional errors or omissions may still exist. Readers are advised to verify referenced materials on their own. The views and conclusions expressed here are the authors’ own and do not necessarily reflect those of their affiliated organizations. 126 Appendix (List of Tables) 2.1 Concise comparison of Soft sets, HyperSoft sets, and SuperHyperSoft sets. . . . . 10 2.2 Concise comparison between a SuperHyperSoft set and an(m, n)-SuperHyperSoft set on a universeU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Soft Set vs. ContraSoft Set (concise comparison) . . . . . . . . . . . . . . . . . . 19 2.4 Concise comparison of classical soft sets and probabilistic soft sets over a finite universeU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Concise comparison between a classical soft set and a D-soft set over a (finite) universeU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Concise comparison between Soft Sets and GraphicSoft Sets over a universeU. . 60 * 127 Bibliography [1] Pradip Kumar Maji, Ranjit Biswas, and A Ranjan Roy. Soft set theory.Computers & mathematics with applications, 45(4-5):555–562, 2003. [2] Dmitriy Molodtsov. Soft set theory-first results.Computers & mathematics with applications, 37(4- 5):19–31, 1999. [3] Thomas Jech.Set theory: The third millennium edition, revised and expanded. Springer, 2003. [4] Lotfi A Zadeh. Fuzzy sets.Information and control, 8(3):338–353, 1965. [5] Krassimir T Atanassov. Circular intuitionistic fuzzy sets.Journal of Intelligent & Fuzzy Systems, 39(5):5981–5986, 2020. [6] Vicenç Torra. Hesitant fuzzy sets.International journal of intelligent systems, 25(6):529–539, 2010. [7] Bui Cong Cuong. Picture fuzzy sets.Journal of Computer Science and Cybernetics, 30:409, 2015. [8] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Single valued neutrosophic graphs.Journal of New theory, 10:86–101, 2016. [9] Haibin Wang, Florentin Smarandache, Yanqing Zhang, and Rajshekhar Sunderraman.Single valued neu- trosophic sets. Infinite study, 2010. [10] R Radha, A Stanis Arul Mary, and Florentin Smarandache. Quadripartitioned neutrosophic pythagorean soft set.International Journal of Neutrosophic Science (IJNS) Volume 14, 2021, page 11, 2021. [11] Rama Mallick and Surapati Pramanik.Pentapartitioned neutrosophic set and its properties, volume 36. Infinite Study, 2020. [12] Lin Wei. An integrated decision-making framework for blended teaching quality evaluation in college english courses based on the double-valued neutrosophic sets.J. Intell. Fuzzy Syst., 45:3259–3266, 2023. [13] Hu Zhao and Hong-Ying Zhang. On hesitant neutrosophic rough set over two universes and its application. Artificial Intelligence Review, 53:4387–4406, 2020. [14] Florentin Smarandache.Plithogenic set, an extension of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets-revisited. Infinite study, 2018. [15] Florentin Smarandache.Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n- SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra. In- finite Study, 2020. [16] Feng Feng, Xiaoyan Liu, Violeta Leoreanu-Fotea, and Young Bae Jun. Soft sets and soft rough sets. Information Sciences, 181(6):1125–1137, 2011. [17] Lotfi A Zadeh. A note on z-numbers.Information sciences, 181(14):2923–2932, 2011. [18] Florentin Smarandache. A unifying field in logics: Neutrosophic logic. InPhilosophy, pages 1–141. Amer- ican Research Press, 1999. [19] Naeem Jan, Tahir Mahmood, Lemnaouar Zedam, and Zeeshan Ali. Multi-valued picture fuzzy soft sets and their applications in group decision-making problems.Soft Computing, 24:18857 – 18879, 2020. [20] Eugenio Aguirre and Antonio González. Fuzzy behaviors for mobile robot navigation: design, coordination and fusion.International Journal of Approximate Reasoning, 25(3):255–289, 2000. [21] Alberto Fernandez, Francisco Herrera, Oscar Cordon, Maria Jose del Jesus, and Francesco Marcelloni. Evolutionary fuzzy systems for explainable artificial intelligence: Why, when, what for, and where to? IEEE Computational intelligence magazine, 14(1):69–81, 2019. [22] Yasmine M Ibrahim, Reem Essameldin, and Saad M Darwish. An adaptive hate speech detection approach using neutrosophic neural networks for social media forensics.Computers, Materials & Continua, 79(1), 2024. [23] OM Khaled, A Salama, Mostafa Herajy, M El-Kirany, Huda E Khalid, Ahmed K Essa, and Ramiz Sabbagh. A novel approach for cyber-attack detection in iot networks with neutrosophic neural networks. Neutrosophic Sets and Systems, 86(1):48, 2025. [24] Florentin Smarandache.New types of soft sets “hypersoft set, indetermsoft set, indetermhypersoft set, and treesoft set”: an improved version. Infinite Study, 2023. 129 Bibliography [25] Muhammad Ihsan, Atiqe Ur Rahman, and Muhammad Haris Saeed. Hypersoft expert set with application in decision making for recruitment process. InNeutrosophic Sets and Systems, 2021. [26] Florentin Smarandache. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic sets and systems, 22(1):168–170, 2018. [27] Oswaldo Edison García Brito, Andrea Sofía Ribadeneira Vacacela, Carmen Hortensia Sánchez Burneo, and Mónica Cecilia Jimbo Galarza. English for specific purposes in the medical sciences to strengthen the pro- fessional profile of the higher education medicine student: a knowledge representation using superhypersoft sets.Neutrosophic Sets and Systems, 74(1):10, 2024. [28] Mona Mohamed, Alaa Elmor, Florentin Smarandache, and Ahmed A Metwaly. An efficient superhypersoft framework for evaluating llms-based secure blockchain platforms.Neutrosophic Sets and Systems, 72:1–21, 2024. [29] T Kiruthika, M Karpagadevi, S Krishnaprakash, and G Deepa. Superhypersoft sets using python and its applications in neutrosophic superhypersoft sets under topsis method.Neutrosophic Sets and Systems, 91:586–616, 2025. [30] Florentin Smarandache. Foundation of the superhypersoft set and the fuzzy extension superhypersoft set: A new vision.Neutrosophic Systems with Applications, 11:48–51, 2023. [31] Ali Alqazzaz and Karam M Sallam. Evaluation of sustainable waste valorization using treesoft set with neutrosophic sets.Neutrosophic Sets and Systems, 65(1):9, 2024. [32] Edwin Collazos Paucar, Jeri G Ramón Ruffner de Vega, Efrén S Michue Salguedo, Agustina C Torres- Rodríguez, and Patricio A Santiago-Saturnino. Analysis using treesoft set of the strategic development plan for extreme poverty municipalities.Neutrosophic Sets and Systems, 69(1):3, 2024. [33] G Dhanalakshmi, S Sandhiya, Florentin Smarandache, et al. Selection of the best process for desalination under a treesoft set environment using the multi-criteria decision-making method.International Journal of Neutrosophic Science, 23(3):140–40, 2024. [34] Mona Gharib, Fatima Rajab, and Mona Mohamed. Harnessing tree soft set and soft computing techniques’ capabilities in bioinformatics: Analysis, improvements, and applications.Neutrosophic sets and systems, 61:579–597, 2023. [35] Takaaki Fujita. Polytree-soft sets and polyforest-soft sets: A directed acyclic framework for soft set modeling.HyperSoft Set Methods in Engineering, 4:11–23, 2025. [36] Takaaki Fujita, Arif Mehmood, Ajoy Kanti Das, Suman Das, Volkan Duran, Arkan A Ghaib, and Ta- lal Al-Hawary. Multitree-soft, pseudotree-soft set, hypertree-soft, andtree-to-tree-soft set.Neutrosophic Computing and Machine Learning. ISSN 2574-1101, 42:65–87, 2026. [37] Florentin Smarandache. New types of soft sets: Hypersoft set, indetermsoft set, indetermhypersoft set, and treesoft set.International Journal of Neutrosophic Science, 2023. [38] Hairong Luo. Forestsoft set approach for estimating innovation and entrepreneurship education in univer- sities through a hierarchical and uncertainty-aware analytical framework.Neutrosophic Sets and Systems, 86(1):21, 2025. [39] Takaaki Fujita, Ajoy Kanti Das, Arif Mehmood, Suman Das, and Volkan Duran. Decision analytics applications of the relationship between treesoft graphs and forestsoft graphs.Applied Decision Analytics, 2(1):73–92, 2026. [40] Takaaki Fujita and Florentin Smarandache.Quantum-TreeSoft Set and Quantum-ForestSoft Set. Infinite Study, 2025. [41] P Sathya, Nivetha Martin, and Florentine Smarandache. Plithogenic forest hypersoft sets in plithogenic contradiction based multi-criteria decision making.Neutrosophic Sets and Systems, 73:668–693, 2024. [42] Viviana del Rocío Marfetan Marfetan, Lesly Gissela Tipanguano Chicaiza, Styven Andrés Pila Chicaiza, and Estephany Monserrath Ojeda Sanchez. Classification of cases of animal abuse in ecuador using inde- termsoft and c4. 5 algorithms.Neutrosophic Sets and Systems, 92:121–133, 2025. [43] Erick González Caballero, Ketty Marilú Moscoso-Paucarchuco, Noel Batista Hernandez, Lorenzo Jo- vanny Cevallos Torres, Maikel Leyva, and Victor Gustavo Gómez Rodríguez. Algorithms of designing decision trees from indeterm soft sets. InNeutrosophic and Plithogenic Inventory Models for Applied Mathematics, pages 561–586. IGI Global Scientific Publishing, 2025. [44] Wei Wei and Pingting Peng. Weighted indetermsoft set for prioritized decision-making with indeterminacy and its application to green competitiveness evaluation in equipment manufacturing enterprises.Neutro- sophic Sets and Systems, 85:1018–1026, 2025. [45] Hai Yang and Cuijuan Lin. A recursive indetermtree soft set (rit-soft set) for dynamic and uncertain performance evaluation in college competitive sports.Neutrosophic Sets and Systems, 85:874–886, 2025. [46] Tao Shen and Chunmei Mao. Sustainability impact of online consumption behavior from the perspective of digital empowerment: Indetermsoft set with application.Neutrosophic Sets and Systems, 82(1):24, 2025. [47] Bhargavi Krishnamurthy and Sajjan G Shiva. Indetermsoft-set-based d* extra lite framework for resource provisioning in cloud computing.Algorithms, 17(11):479, 2024. 130 Bibliography [48] Florentin Smarandache.Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. Infinite Study, 2022. [49] Yan Xu. A neutrosophicα-discounting indetermhypersoft framework for evaluating agricultural product export trade quality under uncertainty.Neutrosophic Sets and Systems, 87:533–542, 2025. [50] Lingling Chen. A comprehensive indetermhypersoft set model for evaluating university literature education effectiveness: Integrating cultural context, argumentation skills, and dynamic progress.Neutrosophic Sets and Systems, 87:295–309, 2025. [51] Takaaki Fujita and Florentin Smarandache. An introduction to advanced soft set variants: Superhypersoft sets, indetermsuperhypersoft sets, indetermtreesoft sets, bihypersoft sets, graphicsoft sets, and beyond. Neutrosophic Sets and Systems, 82:817–843, 2025. [52] Takaaki Fujita and Florentin Smarandache.Navigating Bipolar Indeterminacy: Bipolar IndetermSoft Sets and Bipolar IndetermHyperSoft Sets for Knowledge Representation. Infinite Study, 2026. [53] Takaaki Fujita, Raed Hatamleh, and Ahmed Salem Heilat. Contrasoft set and contrarough set with using upside-down logic.Statistics, Optimization & Information Computing, 2025. [54] Vicenç Torra and Yasuo Narukawa. On hesitant fuzzy sets and decision. In2009 IEEE international conference on fuzzy systems, pages 1378–1382. IEEE, 2009. [55] Yiwei Chen, Qiu Xie, Xiaoyu Ma, and Yuwei Li. Optimizing site selection for construction and demo- lition waste resource treatment plants using a hesitant neutrosophic set: a case study in xiamen, china. Engineering Optimization , pages 1–22, 2024. [56] Juanjuan Chen, Shenggang Li, Shengquan Ma, and Xueping Wang. m-polar fuzzy sets: an extension of bipolar fuzzy sets.The scientific world journal, 2014(1):416530, 2014. [57] V Rajam and N Rajesh. Multipolar neutrosophic subalgebras/ideals of up-algebras.International Journal of Neutrosophic Science (IJNS), 23(4), 2024. [58] Muhammad Saqlain, Muhammad Riaz, Natasha Kiran, Poom Kumam, and Miin-Shen Yang. Water quality evaluation using generalized correlation coefficient for m-polar neutrosophic hypersoft sets.Neutrosophic Sets and Systems, vol. 55/2023: An International Journal in Information Science and Engineering, page 58, 2024. [59] M Sivakumar, Rabıyathul Basarıya, Abdul Rajak, M Senthil, T Vetriselvi, G Raja, and R Rajavarman. Transforming arabic text analysis: Integrating applied linguistics with m-polar neutrosophic set mood change and depression on social media.International Journal of Neutrosophic Science (IJNS), 25(2), 2025. [60] Hind Y Saleh, Areen A Salih, Baravan A Asaad, and Ramadhan A Mohammed. Binary bipolar soft points and topology on binary bipolar soft sets with their symmetric properties.Symmetry, 16(1):23, 2023. [61] Asghar Khan, Muhammad Izhar, and Mohammed M. Khalaf. Generalised multi-fuzzy bipolar soft sets and its application in decision making.J. Intell. Fuzzy Syst., 37:2713–2725, 2019. [62] Maha M Saeed, Sagvan Y Musa, Baravan A Asaad, and Zanyar A Ameen. Pythagorean fuzzy n-bipolar soft sets-based multi-criteria decision-making framework for sustainability evaluation and risk assessment in manufacturing industries.Scientific Reports, 15(1):29648, 2025. [63] Sagvan Y. Musa and Baravan A. Asaad. Topological structures via bipolar hypersoft sets.Journal of Mathematics, 2022. [64] Sagvan Y Musa and Baravan A Asaad. Mappings on bipolar hypersoft classes.Neutrosophic Sets and Systems, 53(1):36, 2023. [65] Sagvan Y Musa and Baravan A Asaad. A progressive approach to multi-criteria group decision-making: N-bipolar hypersoft topology perspective.Plos one, 19(5):e0304016, 2024. [66] T. Fujita and A. Mehmood. Extending classical uncertainty models via hyperpolar structures: Fuzzy, neutrosophic, and soft set perspectives.Galoitica: J. Math. Struct. Appl., 12:24–39, 2025. [67] Muhammad Saeed. An introduction to dynamic soft sets: A framework for modeling temporal uncertainty. Available at SSRN 5820784, 2025. [68] Muhammad Saeed, Fatima Razaq, and Muhammad Hassan. Dynamic soft set topology: A novel topological framework incorporating evolving parameter structures, 2025. [69] Muhammad Saeed, Fatima Razaq, Muhammad Hassan, and Dr Atiqe Ur Rahman. Dynamic soft graphs: A unified framework for modeling time-indexed uncertainty in evolving networks, 2025. [70] Himanshukumar R Patel and Vipul A Shah. General type-2 fuzzy logic systems using shadowed sets: a new paradigm towards fault-tolerant control. In2021 Australian & New Zealand Control Conference (ANZCC), pages 116–121. IEEE, 2021. [71] Mohammad Hossein Azadi, Khaled Nawaser, Ali Vafaei-Zadeh, Seyed Najmodin Mousavi, Ra- zieh Bagherzadeh Khodashahri, and Haniruzila Hanifah. Investigating antecedents of customer relationship management using interval type-2 fuzzy fmea approach.International Journal of Business Innovation and Research, 34(2):139–165, 2024. 131 Bibliography [72] Marwan H Hassan, Saad M Darwish, and Saleh M Elkaffas. Type-2 neutrosophic set and their applications in medical databases deadlock resolution.Computers, Materials & Continua, 74(2), 2023. [73] Muslem Al-Saidi, Áron Ballagi, Oday Ali Hassen, and Saad M Saad. Type-2 neutrosophic markov chain model for subject-independent sign language recognition: A new uncertainty–aware soft sensor paradigm. Sensors (Basel, Switzerland), 24(23):7828, 2024. [74] Soumen Kumar Das, F Yu Vincent, Sankar Kumar Roy, and Gerhard Wilhelm Weber. Location–allocation problem for green efficient two-stage vehicle-based logistics system: A type-2 neutrosophic multi-objective modeling approach.Expert Systems with Applications, 238:122174, 2024. [75] Khizar Hayat, Muhammad Irfan Ali, Bing yuan Cao, and Xiaopeng Yang. A new type-2 soft set: Type-2 soft graphs and their applications.Adv. Fuzzy Syst., 2017:6162753:1–6162753:17, 2017. [76] Khizar Hayat, Bing-Yuan Cao, Muhammad Irfan Ali, Faruk Karaaslan, and Zejian Qin. Characterizations of certain types of type 2 soft graphs.Discrete Dynamics in Nature and Society, 2018(1):8535703, 2018. [77] Musavarah Sarwar and Muhammad Akram. Certain hybrid rough models with type-2 soft information. Journal of Multiple-Valued Logic & Soft Computing, 40, 2023. [78] Shumaila Manzoor, Saima Mustafa, Kanza Gulzar, Asim Gulzar, Sadia Nishat Kazmi, Syed Muham- mad Abrar Akber, Rasool Bukhsh, Sheraz Aslam, and Syed Muhammad Mohsin. Multifuzztops: A fuzzy multi-criteria decision-making model using type-2 soft sets and topsis.Symmetry, 16(6):655, 2024. [79] Guzide Senel. Soft topology generated by l-soft sets.Journal of New Theory, 24:88–100, 2018. [80] Arif Mehmood Khattak, Nazia Hanif, Fawad Nadeem, Muhammad Zamir, Choonkil Park, Giorgio Nordo, and Shamoona Jabeen.Soft b-separation axioms in neutrosophic soft topological structures. Infinite Study, 2019. [81] Saleem Abdullah, Imran Khan, and Muhammad Aslam. A new approach to soft set through applications of cubic set.arXiv preprint arXiv:1210.6517, 2012. [82] Srinivasan Vijayabalaji and Kaliyaperumal Punniyamoorthy. Cubic inverse soft set. InSoft Computing, pages 87–94. CRC Press, 2023. [83] G Muhiuddin and Abdullah M Al-roqi. Cubic soft sets with applications in bck/bci-algebras.Annals of Fuzzy Mathematics and Informatics, 8(2):291–304, 2014. [84] Fatia Fatimah, Dedi Rosadi, RB Fajriya Hakim, and José Carlos R. Alcantud. Probabilistic soft sets and dual probabilistic soft sets in decision-making.Neural Computing and Applications, 31:397–407, 2019. [85] Ping Zhu and Qiaoyan Wen. Probabilistic soft sets. In2010 IEEE international conference on granular computing, pages 635–638. IEEE, 2010. [86] Bindu Nila and Jagannath Roy. Analysis of critical success factors of logistics 4.0 using d-number based pythagorean fuzzy dematel method.Decision Making Advances, 2(1):92–104, 2024. [87] Yuzhen Li and Yabin Shao. Fuzzy cognitive maps based on d-number theory.IEEE Access, 10:72702–72716, 2022. [88] Nuttapong Wattanasiripong, Nuchanat Tiprachot, and Somsak Lekkoksung. On tripolar complex fuzzy sets and their application in ordered semigroups.International Journal of Analysis and Applications, 23:139–139, 2025. [89] Songsong Dai. Linguistic complex fuzzy sets.Axioms, 12(4):328, 2023. [90] Faisal Al-Sharqi, Ashraf Al-Quran, et al. Similarity measures on interval-complex neutrosophic soft sets with applications to decision making and medical diagnosis under uncertainty.Neutrosophic Sets and Systems, 51:495–515, 2022. [91] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Complex neutrosophic graphs of type.Collected Papers. Volume VI: On Neutrosophic Theory and Applications, page 204, 2022. [92] Naveed Yaqoob and Muhammad Akram.Complex neutrosophic graphs. Infinite Study, 2018. [93] Tahir Mahmood and Ubaid ur Rehman. A novel approach towards bipolar complex fuzzy sets and their applications in generalized similarity measures.International Journal of Intelligent Systems, 37:535 – 567, 2021. [94] Daniel Ramot, Menahem Friedman, Gideon Langholz, Ron Milo, and Abraham Kandel. On complex fuzzy sets.10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297), 3:1160–1163 vol.2, 2001. [95] Güzide Şenel. A new construction of spheres via soft real numbers and soft points.Mathematics Letters, 4(3):39–43, 2018. [96] Sujoy Das and SK Samanta. On soft complex sets and soft complex numbers.J. fuzzy math, 21(1):195–216, 2013. [97] Sujoy Das and SK Samanta. Soft real sets, soft real numbers and their properties.J. fuzzy Math, 20(3):551–576, 2012. [98] Seok Zun Song, Hee Sik Kim, and Young Bae Jun. Ideal theory in semigroups based on intersectional soft sets.The Scientific World Journal, 2014(1):136424, 2014. 132 Bibliography [99] Eun Hwan Roh and Young Bae Jun. Positive implicative ideals of bck-algebras based on intersectional soft sets.Journal of Applied Mathematics, 2013(1):853907, 2013. [100] G Muhiuddin. Intersectional soft sets theory applied to generalized hypervector spaces.Analele ştiinţifice ale Universităţi” Ovidius” Constanţa. Seria Matematică, 28(3):171–191, 2020. [101] Young Bae Jun, Chul Hwan Park, and Noura Omair Alshehri. Hypervector spaces based on intersectional soft sets. InAbstract and Applied Analysis. Wiley Online Library, 2014. [102] Hüseyin Kamac and Subramanian Petchimuthu. Bipolar n-soft set theory with applications.Soft Com- puting, 24:16727 – 16743, 2020. [103] Muhammad Akram, Arooj Adeel, and José Carlos Rodriguez Alcantud. Group decision-making methods based on hesitant n-soft sets.Expert Syst. Appl., 115:95–105, 2019. [104] Fatia Fatimah and José Carlos Rodriguez Alcantud. The multi-fuzzy n-soft set and its applications to decision-making.Neural Computing and Applications, 33:11437 – 11446, 2021. [105] Fatia Fatimah, Fatia Fatimah, Dedi Rosadi, R. B. Fajriya Hakim, and José Carlos Rodriguez Alcantud. N-soft sets and their decision making algorithms.Soft Computing, 22:3829 – 3842, 2017. [106] Sagvan Y Musa, Ramadhan A Mohammed, and Baravan A Asaad. N-hypersoft sets: An innovative extension of hypersoft sets and their applications.Symmetry, 15(9):1795, 2023. [107] Sagvan Y Musa. N-bipolar hypersoft sets: Enhancing decision-making algorithms.Plos one, 19(1):e0296396, 2024. [108] Orhan Dalkılıç. Unifying relationships in uncertain environments: examining relations in binary soft sets for expressing inter-object correspondence.The Journal of Supercomputing, 81(16):1–29, 2025. [109] Ahu Açıkgöz and Nihal Tas. Binary soft set theory.EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 9(4):452–463, 2016. [110] Muhammad Saqlain, Poom Kumam, and Wiyada Kumam. Multi-criteria decision-making method based on weighted and geometric aggregate operators of linguistic fuzzy-valued hypersoft set with application. Journal of Fuzzy Extension and Applications, 6(2):344–370, 2025. [111] Muhammad Saqlain, Poom Kumam, and Wiyada Kumam. Linguistic hypersoft set with application to multi-criteria decision-making to enhance rural health services.Neutrosophic Sets and Systems, 61:28–52, 2023. [112] Srinivasan Vijayabalaji and Adhimoolam Ramesh. Uncertain multiplicative linguistic soft sets and their application to group decision making.Journal of Intelligent & Fuzzy Systems, 35(3):3883–3893, 2018. [113] Hongjun Guan, Shuang Guan, and Aiwu Zhao. Intuitionistic fuzzy linguistic soft sets and their application in multi-attribute decision-making.Journal of Intelligent & Fuzzy Systems, 31(6):2869–2879, 2016. [114] Zhao Aiwu and Guan Hongjun. Fuzzy-valued linguistic soft set theory and multi-attribute decision-making application.Chaos, Solitons & Fractals, 89:2–7, 2016. [115] Zhifu Tao, Huayou Chen, Ligang Zhou, and Jinpei Liu. 2-tuple linguistic soft set and its application to group decision making.Soft computing, 19:1201–1213, 2015. [116] NLA Mohd Kamal, Lazim Abdullah, and Ilyani Abdullah. Multi-valued neutrosophic linguistic soft set and its application in multi-criteria decision-making.Journal of Advanced Research in Dynamical and Control Systems, 11:12, 2019. [117] Takaaki Fujita. Metafuzzy, metaneutrosophic, metasoft, and metarough set. 2025. [118] Takaaki Fujita. Metastructure, meta-hyperstructure, and meta-superhyper structure.Journal of Comput- ers and Applications, 1(1):1–22, 2025. [119] Asghar Khan, Muhammad Izhar, and Mohammed M Khalaf. Double-framed soft la-semigroups.Journal of Intelligent & Fuzzy Systems, 33(6):3339–3353, 2017. [120] Yanbin Liu, Peina Liang, and Jingjie Ma. An empirical study on the quality of industry-linked education in vocational colleges: Double-framed treesoft set framework.Neutrosophic Sets and Systems, 85(1):32, 2025. [121] Muhammad Saeed, Hafiz Inam ul Haq, and Mubashir Ali. Extension of double frame soft set to double frame hypersoft set (dfss to dfhss).HyperSoft Set Methods in Engineering, 2:18–27, 2024. [122] Muhammad Izhar, Tariq Mahmood, Asghar Khan, Muhammad Farooq, and Kostaq Hila. Double-framed soft set theory applied to abel-grassmann’s hypergroupoids.New Mathematics and Natural Computation, 18(03):819–841, 2022. [123] Muhammad Saeed, Muhammad Rayees Ahmad, Muhammad Saqlain, and Muhammad Riaz. Rudiments of n-framed soft sets.Punjab University Journal of Mathematics, 52(5), 2020. [124] Muhammad Rayees Ahmad, Usman Afzal, Nadir Omer, Ali Delham Algarni, Sara A Ghorashi, and Huda Eltayeb. A computational diagnostic model for infectious diseases via similarity measures on n-framed plithogenic hypersoft sets.Alexandria Engineering Journal, 127:1209–1219, 2025. [125] Usman Afzal, Muhammad Rayees Ahmad, Nazek Alessa, Nauman Raza, Fathea MO Birkea, Salem Alkha- laf, and Nader Omer. Intelligent faculty evaluation and ranking system based on n-framed plithogenic fuzzy hypersoft set and extended nr-topsis.Alexandria Engineering Journal, 109:18–28, 2024. 133 Bibliography [126] Ajoy Kanti Das, Florentin Smarandache, Rakhal Das, and Suman Das. A comprehensive study on decision- making algorithms in retail and project management using double framed hypersoft sets.HyperSoft Set Methods in Engineering, 2:62–71, 2024. [127] Minyan Chen. Double framed hypersoft set for studies factors that influence and ways to improve vocational college instruction in innovation and entrepreneurship.Neutrosophic Sets and Systems, 85(1):5, 2025. [128] Lingling Chen. Measuring teaching success in college foreign literature programs: An evaluation perspective using double framed superhypersoft set.Neutrosophic Sets and Systems, 85(1):7, 2025. [129] Takaaki Fujita. Double-framed superhypersoft set and double-framed treesoft set.Advancing Uncer- tain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 71, 2025. [130] Ke Gong, Panpan Wang, and Zhi Xiao. Bijective soft set decision system based parameters reduction under fuzzy environments.Applied Mathematical Modelling, 37(6):4474–4485, 2013. [131] Ke Gong, Zhi Xiao, and Xia Zhang. The bijective soft set with its operations.Comput. Math. Appl., 60:2270–2278, 2010. [132] Varun Kumar Tiwari, Prashant Kumar Jain, and Puneet Tandon. An integrated shannon entropy and top- sis for product design concept evaluation based on bijective soft set.Journal of Intelligent Manufacturing, 30:1645 – 1658, 2017. [133] Atiqe Ur Rahman, Muhammad Saeed, and Abida Hafeez. Theory of bijective hypersoft set with application in decision making.Punjab University Journal of Mathematics, 53(7), 2021. [134] Takaaki Fujita. N-superhypersoft set and bijective superhypersoft set.Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 138, 2025. [135] Muhammad Ihsan, Muhammad Saeed, Atiqe Ur Rahman, and Florentin Smarandache. Multi-attribute decision support model based on bijective hypersoft expert set.Punjab University Journal of Mathematics, 54(1), 2022. [136] Gustavo Santos-García and José Carlos R Alcantud. Ranked soft sets.Expert Systems, 40(6):e13231, 2023. [137] Irfan Deli. Refined neutrosophic sets and refined neutrosophic soft sets: theory and applications. In Handbook of research on generalized and hybrid set structures and applications for soft computing, pages 321–343. IGI Global, 2016. [138] Takaaki Fujita and Florentin Smarandache.Some types of hyperneutrosophic set (6): Multineutrosophic set and refined neutrosophic set. Infinite Study, 2025. [139] Florentin Smarandache. n-valued refined neutrosophic logic and its applications to physics.Infinite study, 4:143–146, 2013. [140] Anjan Mukherjee, Mithun Datta, and Abhijit Saha. Refined soft sets and its applications.Journal of New Theory, 14:10–25, 2016. [141] Faruk Karaaslan. Correlation coefficients of single-valued neutrosophic refined soft sets and their applica- tions in clustering analysis.Neural Computing and Applications, 28(9):2781–2793, 2017. [142] R Anitha Cruz. Neutrosophic soft cubic refined sets.Neutrosophic Sets & Systems, 73, 2024. [143] Fujita Takaaki and Arif Mehmood. Iterative multifuzzy set, iterative multineutrosophic set, iterative multisoft set, and multiplithogenic sets.Neutrosophic Computing and Machine Learning, 41:1–30, 2025. [144] Shawkat Alkhazaleh, Abdul Razak Salleh, Nasruddin Hassan, and Abd Ghafur Ahmad. Multisoft sets. In Proc. 2nd International Conference on Mathematical Sciences, pages 910–917, 2010. [145] Florentin Smarandache.Practical applications of IndetermSoft Set and IndetermHyperSoft Set and intro- duction to TreeSoft Set as an extension of the MultiSoft Set. Infinite Study, 2022. [146] Sabu Sebastian and TV Ramakrishnan. Multi-fuzzy sets: An extension of fuzzy sets.Fuzzy Information and Engineering, 3:35–43, 2011. [147] Mahalakshmi Pethaperumal, Vimala Jeyakumar, Jeevitha Kannan, and Ashma Banu. An algebraic analy- sis on exploring q-rung orthopair multi-fuzzy sets.Journal of fuzzy extension and applications, 4(3):235–245, 2023. [148] Johanna Estefanía Street Imbaquingo Street, Karen Milagros Díaz Street Salambay, Jordy Alexis Vargas Yumbo, and Kevin Christopher Carrasco Azogue. Intercultural education from ancestral logic: Applica- tion of the ayni method multi-neutrosophic to strengthen mixed methodology in the waorani community. Neutrosophic Sets and Systems, 92:263–283, 2025. [149] Ennio Jesús Mérida Córdova, Elizabeth Esther Vergel Parejo, and Raúl López Fernández. Scholarai schol- arly article search strategies with the ayni method multi-neutrosophic for ethical information management in ai.Neutrosophic Sets and Systems, 92:380–397, 2025. [150] Takaaki Fujita and Florentin Smarandache. An introduction to advanced soft set variants: Superhypersoft sets, indetermsuperhypersoft sets, indetermtreesoft sets, bihypersoft sets, graphicsoft sets, and beyond. Neutrosophic Sets and Systems, 82:817–843, 2025. [151] Ari M Lipsky and Sander Greenland. Causal directed acyclic graphs.JAMA, 327(11):1083–1084, 2022. 134 Bibliography [152] Takaaki Fujita. Directed acyclic superhypergraphs (dash): A general framework for hierarchical depen- dency modeling.Neutrosophic Knowledge, 6:72–86, 2025. [153] Peter WG Tennant, Eleanor J Murray, Kellyn F Arnold, Laurie Berrie, Matthew P Fox, Sarah C Gadd, Wendy J Harrison, Claire Keeble, Lynsie R Ranker, Johannes Textor, et al. Use of directed acyclic graphs (dags) to identify confounders in applied health research: review and recommendations.International journal of epidemiology, 50(2):620–632, 2021. [154] Weizhou Shen, Siyue Wu, Yunyi Yang, and Xiaojun Quan. Directed acyclic graph network for conversa- tional emotion recognition.arXiv preprint arXiv:2105.12907, 2021. [155] Takaaki Fujita and Florentin Smarandache. An introduction to advanced soft set variants: Superhypersoft sets, indetermsuperhypersoft sets, indetermtreesoft sets, bihypersoft sets, graphicsoft sets, and beyond. Neutrosophic Sets and Systems, 82:817–843, 2025. [156] Mehmet Şahin, İrfan Deli, and Vakkas Uluçay.Bipolar Neutrosophic Soft Expert Sets. Infinite Study, 2016. [157] Faisal Al-Sharqi, Abd Ghafur Ahmad, and Ashraf Al-Quran. Interval-valued neutrosophic soft expert set from real space to complex space.CMES-Computer Modeling in Engineering & Sciences, 132(1), 2022. [158] Ashraf Al-Quran and Nasruddin Hassan. The complex neutrosophic soft expert set and its application in decision making.Journal of Intelligent & Fuzzy Systems, 34(1):569–582, 2018. [159] Ashraf Al-Quran, Nasruddin Hassan, and Shawkat Alkhazaleh. Fuzzy parameterized complex neutrosophic soft expert set for decision under uncertainty.Symmetry, 11(3):382, 2019. [160] Fathima Perveen PA, Sunil Jacob John, et al. On spherical fuzzy soft expert sets. InAIP conference proceedings. AIP Publishing, 2020. [161] Yousef Al-Qudah and Nasruddin Hassan. Fuzzy parameterized complex multi-fuzzy soft expert sets.THE 2018 UKM FST POSTGRADUATE COLLOQUIUM: Proceedings of the Universiti Kebangsaan Malaysia, Faculty of Science and Technology 2018 Postgraduate Colloquium, 2019. [162] Mehmet Sahin, Shawkat Alkhazaleh, and Vakkas Ulucay. Neutrosophic soft expert sets.Applied Mathematics-a Journal of Chinese Universities Series B, 06:116–127, 2015. [163] Faisal Al-Sharqi, Yousef Al-Qudah, and Naif Alotaibi. Decision-making techniques based on similarity measures of possibility neutrosophic soft expert sets.Neutrosophic Sets and Systems, vol. 55/2023: An International Journal in Information Science and Engineering, page 358, 2024. [164] Sumyyah Al-Hijjawi, Abd Ghafur Ahmad, and Shawkat Alkhazaleh. Effective neutrosophic soft expert set and its application.International Journal of Neutrosophic Science (IJNS), 23(1), 2024. [165] Takaaki Fujita. Superhypersoft rough set, superhypersoft expert set, and bipolar superhypersoft set. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 270, 2025. [166] Ashraf Al-Quran, Nasruddin Hassan, and Emad A. Marei. A novel approach to neutrosophic soft rough set under uncertainty.Symmetry, 11:384, 2019. [167] Xinyi Wang and Qinghai Wang. Uncertainty measurement of variable precision fuzzy soft rough set model. InCECNet, 2022. [168] Tasawar Abbas, Rehan Zafar, Sana Anjum, Ambreen Ayub, and Zamir Hussain. An innovative soft rough dual hesitant fuzzy sets and dual hesitant fuzzy soft rough sets.VFAST Transactions on Mathematics, 2023. [169] Fu Zhang, Weimin Ma, and Hongwei Ma. Dynamic chaotic multi-attribute group decision making under weighted t-spherical fuzzy soft rough sets.Symmetry, 15:307, 2023. [170] Aysun Benek and Taha Yasin Ozturk. A comparative analysis of two different decision-making methods in neutrosophic soft rough set environments.OPSEARCH, pages 1–22, 2025. [171] Jingjing Zhang. Neutrosophic soft rough sets for quality evaluation of interactive music teaching in higher education: A novel approach.Neutrosophic Sets and Systems, 90(1):66, 2025. [172] Siyang Yang. Extending superhypersoft framework: Weighted soft sets for priority-based decision-making in engineering ethics risk analysis based on big data technology.Neutrosophic Sets and Systems, 86:119–125, 2025. [173] K Selvakumari. Solving game problem using weighted soft sets.Journal of Computer and Mathematical Sciences, 9(10):1307–1311, 2018. [174] Holy-Heavy M Balami, Aliyu G Dzarma, and Mohammed A Mohammed. Weighted soft set and its application in parameterized decision making processes.International Journal of Development Mathematics (IJDM), 2(1):131–144, 2025. [175] Omer Akguller. Geometric soft sets.Hittite Journal of Science and Engineering, 4(2):159–164, 2017. [176] Abdul Razak Salleh, Shawkat Alkhazaleh, Nasruddin Hassan, and Abd Ghafur Ahmad. Multiparameter- ized soft set.Journal of Mathematics and Statistics, 8(1):92–97, 2012. [177] Takaaki Fujita and Iqbal M Batiha. Multiparameterized hypersoft set and type-2 hypersoft set.Neutro- sophic Sets and Systems, 95:183–199, 2026. 135 Bibliography [178] Young Bae Jun, Seok Zun Song, and G Muhiuddin. Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups.The Scientific World Journal, 2014(1):467968, 2014. [179] Atiqe Ur Rahman, Muhammad Saeed, and Florentin Smarandache.Convex and concave hypersoft sets with some properties, volume 38. Infinite Study, 2020. [180] İrfan Deli. Convex and concave sets based on soft sets and fuzzy soft sets.Journal of New Theory, 29:101–110, 2019. [181] P. A. Fathima Perveen and Sunil Jacob John. Relations on spherical fuzzy soft sets.2nd INTERNATIONAL CONFERENCE ON COMPUTATIONAL SCIENCES-MODELLING, COMPUTING AND SOFT COM- PUTING (CSMCS 2022), 2023. [182] Sujit Das and Samarjit Kar. Intuitionistic multi fuzzy soft set and its application in decision making. In Pattern Recognition and Machine Intelligence: 5th International Conference, PReMI 2013, Kolkata, India, December 10-14, 2013. Proceedings 5, pages 587–592. Springer, 2013. [183] Muhammad Saeed, Irfan Saif Ud Din, Imtiaz Tariq, and Harish Garg. Refined fuzzy soft sets: Properties, set-theoretic operations and axiomatic results.Journal of Computational and Cognitive Engineering, 3(1):24–33, 2024. [184] Sheikh Zain Majid, Muhammad Saeed, Umar Ishtiaq, and Ioannis K Argyros. The development of a hybrid model for dam site selection using a fuzzy hypersoft set and a plithogenic multipolar fuzzy hypersoft set. Foundations, 4(1):32–46, 2024. [185] Xingsi Xue, Himanshu Dhumras, Garima Thakur, Rakesh Kumar Bajaj, and Varun Shukla. Schweizer- sklar t-norm operators for picture fuzzy hypersoft sets: Advancing suistainable technology in social healthy environments.Computers, Materials & Continua, 84(1), 2025. [186] R Hema, R Sudharani, and M Kavitha. A novel approach on plithogenic interval valued neutrosophic hypersoft sets and its application in decision making.Indian Journal Of Science And Technology, 2023. [187] Takaaki Fujita. Hyperfuzzy hypersoft set and hyperneutrosophic hypersoft set.Advancing Uncertain Com- binatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 247, 2025. [188] Francina Shalini. Trigonometric similarity measures of pythagorean neutrosophic hypersoft sets.Neutro- sophic Systems with Applications, 2023. [189] Muhammad Saqlain and Xiao Long Xin.Interval valued, m-polar and m-polar interval valued neutrosophic hypersoft sets. Infinite Study, 2020. [190] Yuncheng Jiang, Yong Tang, Qimai Chen, Hai Liu, and Jianchao Tang. Interval-valued intuitionistic fuzzy soft sets and their properties.Computers & Mathematics with Applications, 60(3):906–918, 2010. [191] Harish Garg and Rishu Arora. Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making.Journal of the Operational Research Society, 69:1711 – 1724, 2018. [192] Harish Garg and Rishu Arora. Topsis method based on correlation coefficient for solving decision-making problems with intuitionistic fuzzy soft set information. InAIMS mathematics, 2020. [193] Harish Garg and Rishu Arora. Generalized maclaurin symmetric mean aggregation operators based on archimedean t-norm of the intuitionistic fuzzy soft set information.Artificial Intelligence Review, 54:3173 – 3213, 2020. [194] Krassimir T Atanassov and G Gargov.Intuitionistic fuzzy logics. Springer, 2017. [195] Shawkat Alkhazaleh. n-valued refined neutrosophic soft set theory.Journal of Intelligent & Fuzzy Systems, 32(6):4311–4318, 2017. [196] Shawkat Alkhazaleh and Ayman A Hazaymeh. N-valued refined neutrosophic soft sets and their appli- cations in decision making problems and medical diagnosis.Journal of Artificial Intelligence and Soft Computing Research, 8(1):79–86, 2018. [197] Muhammad Akram and Sundas Shahzadi.Representation of graphs using intuitionistic neutrosophic soft sets. Infinite Study, 2016. [198] S Broumi and Tomasz Witczak. Heptapartitioned neutrosophic soft set.International Journal of Neutro- sophic Science, 18(4):270–290, 2022. [199] Quang-Thinh Bui, My-Phuong Ngo, Vaclav Snasel, Witold Pedrycz, and Bay Vo. The sequence of neu- trosophic soft sets and a decision-making problem in medical diagnosis.International Journal of Fuzzy Systems, 24:2036 – 2053, 2022. [200] Hüseyin Kamacı. Linguistic single-valued neutrosophic soft sets with applications in game theory.Inter- national Journal of Intelligent Systems, 36(8):3917–3960, 2021. [201] S. Onar. A note on neutrosophic soft set over hyperalgebras.Symmetry, 16(10):1288, 2024. [202] Faruk Karaaslan.Neutrosophic soft sets with applications in decision making. Infinite Study, 2014. [203] Pabitra Kumar Maji.Neutrosophic soft set. Infinite Study, 2013. 136 Bibliography [204] Fazeelat Sultana, Muhammad Gulistan, Mumtaz Ali, Naveed Yaqoob, Muhammad Khan, Tabasam Rashid, and Tauseef Ahmed. A study of plithogenic graphs: applications in spreading coronavirus disease (covid-19) globally.Journal of ambient intelligence and humanized computing, 14(10):13139–13159, 2023. [205] Nivetha Martin. Introduction to possibility plithogenic soft sets.Plithogenic Logic and Computation, 2024. [206] Shawkat Alkhazaleh.Plithogenic soft set. Infinite Study, 2020. [207] Takaaki Fujita and Florentin Smarandache. A unified framework foru-structures and functorial structure: Managing super, hyper, superhyper, tree, and forest uncertain over/under/off models.Neutrosophic Sets and Systems, 91:337–380, 2025. [208] Takaaki Fujita and Florentin Smarandache.HyperGraph and SuperHyperGraph Theory with Applications (IV): Uncertain Graph Theory, volume IV ofHyperGraph and SuperHyperGraph Theory with Applications. Neutrosophic Science International Association (NSIA) Publishing House, 1.0 edition, 2026. [209] Takaaki Fujita and Florentin Smarandache.HyperGraph and SuperHyperGraph Theory with Applications. Neutrosophic Science International Association (NSIA) Publishing House, 2026. [210] YS Yun. Parametric operations between 3-dimensional triangular fuzzy number and trapezoidal fuzzy set. Journal of Algebra & Applied Mathematics, 21(2), 2023. [211] Tong Shaocheng. Interval number and fuzzy number linear programmings.Fuzzy sets and systems, 66(3):301–306, 1994. [212] Takaaki Fujita and Florentin Smarandache.A Dynamic Survey of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, Plithogenic, and Extensional Sets. Neutrosophic Science International Association (NSIA), 2025. [213] Jyoti D Thenge, B Surendranath Reddy, and Rupali S Jain. Contribution to soft graph and soft tree.New Mathematics and Natural Computation, 15(01):129–143, 2019. [214] Muhammad Akram and Saira Nawaz. Operations on soft graphs.Fuzzy information and Engineering, 7(4):423–449, 2015. [215] Muhammad Saeed, Muhammad Khubab Siddique, Muhammad Ahsan, Muhammad Rayees Ahmad, and Atiqe Ur Rahman. A novel approach to the rudiments of hypersoft graphs.Theory and Application of Hypersoft Set, Pons Publication House, Brussel, pages 203–214, 2021. [216] Muhammad Saeed, Atiqe Ur Rahman, and Muhammad Arshad. A study on some operations and products of neutrosophic hypersoft graphs.Journal of Applied Mathematics and Computing, 68(4):2187–2214, 2022. [217] Muhammad Saeed, Muhammad Imran Harl, Muhammad Haris Saeed, and Ibrahim Mekawy. Theoretical framework for a decision support system for micro-enterprise supermarket investment risk assessment using novel picture fuzzy hypersoft graph.Plos one, 18(3):e0273642, 2023. [218] R. Jahir Hussain and M. S. Afya Farhana. Fuzzy chromatic number of fuzzy soft cycle and complete fuzzy soft graphs.AIP Conference Proceedings, 2023. [219] Umair Amin, Aliya Fahmi, Yaqoob Naveed, Aqsa Farid, and Muhammad Arshad Shehzad Hassan. Domi- nation in bipolar fuzzy soft graphs.J. Intell. Fuzzy Syst., 46:6369–6382, 2024. [220] Vakkas Ulucay. Q-neutrosophic soft graphs in operations management and communication network.Soft Computing, 25:8441 – 8459, 2021. [221] S Satham Hussain, R Hussain, and Florentin Smarandache. Domination number in neutrosophic soft graphs.Neutrosophic Sets and Systems, 28:228–244, 2019. [222] Muhammad Akram and Hafiza Saba Nawaz. Implementation of single-valued neutrosophic soft hyper- graphs on human nervous system.Artificial Intelligence Review, 56(2):1387–1425, 2023. [223] Bobin George, Jinta Jose, and Rajesh K Thumbakara. Exploring soft hypergraphs through various oper- ations.New Mathematics and Natural Computation, 20(02):551–566, 2024. [224] Takaaki Fujita, Atiqe Ur Rahman, Arkan A Ghaib, Talal Ali Al-Hawary, and Arif Mehmood Khattak. On the properties and illustrative examples of soft superhypergraphs and rough superhypergraphs.Prospects for Applied Mathematics and Data Analysis, 5(1):12–31, 2025. [225] Jinta Jose, Bobin George, and Rajesh K Thumbakara. Advancements in soft directed graph theory: new ideas and properties.New Mathematics and Natural Computation, pages 1–17, 2024. [226] Jinta Jose, Bobin George, and Rajesh K Thumbakara. Soft directed graphs, their vertex degrees, associated matrices and some product operations.New Mathematics and Natural Computation, 19(03):651–686, 2023. [227] Raed Hatamleh, Nasir Odat, Hamza Ali Abujabal, Faria Khan, Arif Mehmood Khattak, Alaa M. Abd El-latif, Husham M. Attaalfadeel, and Abdelhalim Hasnaoui. Fermatean double-valued neutrosophic soft topological spaces.European Journal of Pure and Applied Mathematics, 2025. [228] Maha Mohammed Saeed, Sami Ullah Khan, Fatima Suriyya, Arif Mehmood, and Jamil J Hamja. Interval- valued complex neutrosophic sets and complex neutrosophic soft topological spaces.International Journal of Analysis and Applications, 23:132–132, 2025. [229] V Subash and M Angayarkanni. Neutrosophic hypersoft topological spaces viam-open sets.JP Journal of Geometry and Topology, 31(1):39–54, 2025. [230] V Subash and M Angayarkanni. Contra m-continuous maps and contra m-irresolute maps in fuzzy hypersoft topological spaces.International Journal of Environmental Sciences, 11(6s):431–448, 2025. 137 Bibliography [231] Sagvan Younis Musa and Baravan Abdulmuhsen Asaad. Connectedness on bipolar hypersoft topological spaces.Journal of Intelligent & Fuzzy Systems, 43(4):4095–4105, 2022. [232] PG Patil, C Jaya Subba Reddy, Rani Teli, and Vyshakha Elluru. New structures in fuzzy binary soft topological spaces.International Journal of Mathematics Trends and Technology-IJMTT, 71, 2025. [233] Rui Gao and Jianrong Wu. Filter with its applications in fuzzy soft topological spaces.AIMS Mathematics, 6(3):2359–2368, 2021. [234] A Mukherjee and AK Das. Parameterized topological space induced by an intuitionistic fuzzy soft multi topological space.Ann. Pure and Applied Math, 7:7–12, 2014. [235] Francisco Gallego Lupiáñez. On intuitionistic fuzzy topological spaces.Kybernetes, 35(5):743–747, 2006. [236] M Parimala, M Karthika, and Florentin Smarandache.A review of fuzzy soft topological spaces, intuition- istic fuzzy soft topological spaces and neutrosophic soft topological spaces. Infinite Study, 2020. [237] Maha Mohammed Saeed, Raed Hatamleh Hatamleh, Alaa M Abd El-latif, Abdallah Al-Husban, Takaaki Fujita, Cris L Armada, Rabia Andleeb, and Arif Mehmood Khattak. Separation axioms in quadri-partition neutrosophic soft topological spaces.European Journal of Pure and Applied Mathematics, 18(3):6324–6324, 2025. [238] S Kumar, A Mary, and R Radha. Penta partitioned neutrosophic soft topological space.Fuzzy, Intuition- istic and Neutrosophic Set Theories and their Applications in Decision Analysis, pages 49–59, 2025. [239] Noori F Al-Mayahi. Soft banach algebra: Theory and applications.Journal of Iraqi Al-Khawarizmi, 8(2):44–68, 2024. [240] Young Bae Jun. Union-soft sets with applications in bck/bci-algebras.Bulletin of the Korean Mathematical Society, 50(6):1937–1956, 2013. [241] Zanyar A Ameen, Tareq M Al-shami, Radwan Abu-Gdairi, and Abdelwaheb Mhemdi. The relationship between ordinary and soft algebras with an application.Mathematics, 11(9):2035, 2023. [242] Nenad Stojanović. Soft sets whose soft measure is zero.Filomat, 39(17):5825–5832, 2025. [243] Vakkas Ulucay, Mehmet Sahin, Necati Olgun, and Adem Klcman. On neutrosophic soft lattices.Afrika Matematika, 28:379–388, 2017. [244] S. Rajareega, J. Felicita Vimala, and D. Preethi. Complex intuitionistic fuzzy soft lattice ordered group and its weighted distance measures.Mathematics, 2020. [245] VD Jobish, KV Babitha, and Sunil Jacob John. On soft lattice operations.J Adv Res Pure Math, 5(2):71–86, 2013. [246] Yingchao Shao and Keyun Qin. Fuzzy soft sets and fuzzy soft lattices.International Journal of Compu- tational Intelligence Systems, 5(6):1135–1147, 2012. [247] Vassilios Petridis and Vassilis G Kaburlasos. Learning in the framework of fuzzy lattices.IEEE Transactions on Fuzzy Systems, 7(4):422–440, 2002. [248] Shio Gai Quek, Ganeshsree Selvachandran, Vimala Jayakumar, Phet Duong, and Le Hoang Son. A new decision making model based on complex intuitionistic fuzzy soft lattice for traffic monitoring in the pandemic scenarios.Advanced Intelligent Systems, 6(11):2400145, 2024. [249] S Rajareega, J Vimala, and D Preethi. Complex intuitionistic fuzzy soft lattice ordered group and its weighted distance measures.Mathematics, 8(5):705, 2020. [250] S Rajareega and J Vimala. Operations on complex intuitionistic fuzzy soft lattice ordered group and cifs- copras method for equipment selection process.Journal of Intelligent & Fuzzy Systems, 41(5):5709–5718, 2021. [251] A Sezgin Sezer and AO Atagün. A new kind of vector space: soft vector space.Southeast asian bulletin of mathematics, 40(5):753–770, 2016. [252] C Gunduz Aras, AYSE Sonmez, and HUSEYIN Cakalli. An approach to soft functions.J. Math. Anal, 8(2):129–138, 2017. [253] Sabir Hussain. On some soft functions.Mathematical Sciences Letters, 4(1):55, 2015. [254] Zanyar A Ameen and Mesfer H Alqahtani. Some classes of soft functions defined by soft open sets modulo soft sets of the first category.Mathematics, 11(20):4368, 2023. [255] Hacı Aktaş and Naim Çağman. Soft sets and soft groups.Information sciences, 177(13):2726–2735, 2007. [256] Ajoy Kanti Das and Carlos Granados. An advanced approach to fuzzy soft group decision-making using weighted average ratings.SN Computer Science, 2(6):471, 2021. [257] Muhammad Saeed, Atiqe Ur Rahman, Muhammad Ahsan, and Florentin Smarandache. An inclusive study on fundamentals of hypersoft set.Theory and Application of Hypersoft Set, 1:1–23, 2021. [258] Abdülkadir Aygünolu and Halis Aygün. Introduction to fuzzy soft groups.Computers & Mathematics with Applications, 58(6):1279–1286, 2009. [259] Majdoleen Abu Qamar and Nasruddin Hassan.Characterizations of group theory under Q-neutrosophic soft environment. Infinite Study, 2019. 138 Bibliography [260] Yıldıray Celik, Canan Ekiz, and Sultan Yamak. Applications of fuzzy soft sets in ring theory.Annals Fuzzy Mathematics and Informatics, 5(3):451–462, 2013. [261] Jayanta Ghosh, Dhananjoy Mandal, and T Samanta. Soft structures of groups and rings.International Journal of Scientific World, 5(2):117–125, 2017. [262] Ummahan Acar, Fatih Koyuncu, and Bekir Tanay. Soft sets and soft rings.Computers & Mathematics with Applications, 59(11):3458–3463, 2010. [263] Roy Goetschel and William Voxman. Fuzzy matroids. InFuzzy sets and systems, 1988. [264] Roy Goetschel and William Voxman. Bases of fuzzy matroids.Fuzzy Sets and Systems, 31:253–261, 1989. [265] Ladislav A. Novak. On goetschel and voxman fuzzy matroids.Fuzzy Sets Syst., 117:407–412, 2001. [266] Kholod M Hassan and Saied A Johnny. Matroidal structure based on soft-sets. InJournal of Physics: Conference Series. IOP Publishing, 2020. [267] Muhammad Akram, Musavarah Sarwar, and Wieslaw A Dudek. Bipolar fuzzy circuits. InGraphs for the Analysis of Bipolar Fuzzy Information, pages 281–307. Springer, 2020. [268] Ahmed B AL-Nafee, Said Broumi, and Florentin Smarandache.Neutrosophic soft bitopological spaces. Infinite Study, 2021. [269] A Kandil, OAE Tantawy, SA El-Sheikh, and Shawqi A Hazza. Pairwise open (closed) soft sets in soft bitopological spaces.Ann. Fuzzy Math. Inform, 11(4):571–588, 2016. [270] Basavaraj M Ittanagi. Soft bitopological spaces.International Journal of Computer Applications, 107(7):1–4, 2014. [271] AF Sayed. Some separation axioms in fuzzy soft bitopological spaces.J. Math. Comput. Sci., 8(1):28–45, 2017. [272] Taha Yasin Ozturk and Sadi Bayramov. Category of chain complexes of soft modules.International Mathematical Forum, 7(20):981–992, 2012. [273] Mohammed Amare Mohammed, Berehanu Bekele Belayneh, Zelalem Teshome Wale, Gezahagne Mulat Addis, and Mohammed Tesemma. Construction of soft modules over soft abelian groups.Research in Mathematics, 13(1):2605729, 2026. [274] Mikail Bal and Necati Olgun. Soft neutrosophic modules.Mathematics, 6(12):323, 2018. [275] Qiu-Mei Sun, Zi-Long Zhang, and Jing Liu. Soft sets and soft modules. InInternational Conference on Rough Sets and Knowledge Technology, pages 403–409. Springer, 2008. [276] Sadi Bayramov, Cigdem Gunduz, and M Ibrahim Yazar. Inverse system of fuzzy soft modules.Annals of Fuzzy Mathematics and Informatics, 4(2):349–363, 2012. [277] OA Tantawy and RM Hassan. Soft metric spaces. In5th International Conference on Mathematics and Information Sciences, 2016. [278] İsmet Altıntaş and Peyil Esengul kyzy. Topology of soft partial metric spaces.Soft Computing, 29(19):5613–5623, 2025. [279] Vildan Çetkin, Elif Güner, and Halis Aygün. On 2s-metric spaces.Soft Computing, 24(17):12731–12742, 2020. [280] Sonam, Ramakant Bhardwaj, Josika Mal, Pulak Konar, and Phumin Sumalai. Fixed point results in soft probabilistic metric spaces.The journal of Analysis, 33(1):139–166, 2025. [281] Yuan Zou. Bayesian decision making under soft probabilities.Journal of Intelligent & Fuzzy Systems, 44(6):10661–10673, 2023. [282] DA Molodtsov. Soft probability of large deviations.Advances in Systems Science and Applications, 13(1):53–67, 2013. [283] Jing Qiu, Zhi Xiao, Wei Xu, and Ying Zhou. Soft probability based random forest for financial distress prediction.Information Sciences, page 122870, 2025. [284] Trevor Jack. On the complexity of properties of partial bijection semigroups, 2021. [285] Rukchart Prasertpong and Aiyared Iampan. Approximation approaches for rough hypersoft sets based on hesitant bipolar-valued fuzzy hypersoft relations on semigroups.Journal of Mathematics and Computer Science, 2022. [286] Young Bae Jun, Kyoung Ja Lee, and Asghar Khan. Soft ordered semigroups.Mathematical Logic Quarterly, 56(1):42–50, 2010. [287] Tahir Mahmood, Muhammad Asif, Ubaid ur Rehman, and Jabbar Ahmmad. T-bipolar soft semigroups and related results.Spectrum of Mechanical Engineering and Operational Research, 1(1):258–271, 2024. [288] Munazza Naz, Muhammad Shabir, and Muhammad Irfan Ali. On fuzzy soft semigroups.World Applied Sciences Journal (Special Issue of Applied Math), 22:62–83, 2013. [289] Cheng-Fu Yang. Fuzzy soft semigroups and fuzzy soft ideals.Computers & Mathematics with Applications, 61(2):255–261, 2011. [290] M Al Tahan and Bijan Davvaz. Weak chemical hyperstructures associated to electrochemical cells.Iranian Journal of Mathematical Chemistry, 9(1):65–75, 2018. 139 Bibliography [291] Maria Santilli Ruggero and Thomas Vougiouklis. Hyperstructures in lie-santilli admissibility and iso- theories.Ratio Mathematica, 33:151, 2017. [292] Florentin Smarandache. Foundation of superhyperstructure & neutrosophic superhyperstructure.Neutro- sophic Sets and Systems, 63(1):21, 2024. [293] Sultan Yamak, Osman Kazancı, and Bijan Davvaz. Soft hyperstructure.Computers & Mathematics with Applications, 62(2):797–803, 2011. [294] Gulay Oguz and Bijan Davvaz. Soft topological hyperstructure.J. Intell. Fuzzy Syst., 40:8755–8764, 2021. [295] GR Amiri, R Mousarezaei, and S Rahnama. Soft hyperstructures and their applications.New Mathematics and Natural Computation, pages 1–19, 2024. [296] Takaaki Fujita and Florentin Smarandache.Superhypergraph neural networks and plithogenic graph neural networks: Theoretical foundations. Infinite Study, 2025. [297] A Meenakshi, J Shivangi Mishra, Jeong Gon Lee, Antonios Kalampakas, and Sovan Samanta. Advanced risk prediction in healthcare: Neutrosophic graph neural networks for disease transmission.Complex & Intelligent Systems, 11(9):413, 2025. [298] Filip Ekström Kelvinius, Dimitar Georgiev, Artur Toshev, and Johannes Gasteiger. Accelerating molecular graph neural networks via knowledge distillation.Advances in Neural Information Processing Systems, 36:25761–25792, 2023. [299] Daniel Vik, David Pii, Chirag Mudaliar, Mads Nørregaard-Madsen, and Aleksejs Kontijevskis. Perfor- mance and robustness of small molecule retention time prediction with molecular graph neural networks in industrial drug discovery campaigns.Scientific Reports, 14(1):8733, 2024. [300] Yingfang Yuan, Wenjun Wang, Xin Li, Kefan Chen, Yonghan Zhang, and Wei Pang. Evolving molecular graph neural networks with hierarchical evaluation strategy. InProceedings of the Genetic and Evolutionary Computation Conference, pages 1417–1425, 2024. [301] Midhilesh Momidi, Priyanka S Chauhan, Adityaram Komaraneni, Surya Prakash Ghattamaneni, Kamal Upreti, and Nishant Kumar. Uncertainty-aware molecular property prediction using heterogeneous molecu- lar graph neural networks. InInternational Conference on Generative Artificial Intelligence, Cryptography, and Predictive Analytics, pages 243–254. Springer, 2024. [302] A Salama, Huda E Khalid, Ahmed K Essa, and Nadheer M Ahmed. A natural language processing environment for rule-based decision making with neutrosophic logic to manage uncertainty and ambiguity. Neutrosophic Sets and Systems, 82(1):44, 2025. [303] Diego Fernando Coka Flores, Ignacio Fernando Barcos Arias, María Elena Infante Miranda, and Omar Mar Cornelio. Applying neutrosophic natural language processing to analyze complex phenomena in interdis- ciplinary contexts.Neutrosophic Sets and Systems, 74:297–305, 2024. [304] Sultan AlGhozali and Siti Mukminatun. Natural language processing of gemini artificial intelligence pow- ered chatbot.Balangkas: An International Multidisciplinary Research Journal, 1(1):41–48, 2024. [305] Naeemeh Adel.Fuzzy natural language similarity measures through computing with words. PhD thesis, Manchester Metropolitan University, 2022. [306] Yenson Vinicio Mogro Cepeda, Marco Antonio Riofrío Guevara, Emerson Javier Jácome Mogro, and Rachele Piovanelli Tizano. Impact of irrigation water technification on seven directories of the san juan- patoa river using plithogenic n-superhypergraphs based on environmental indicators in the canton of pujilí, 2021.Neutrosophic Sets and Systems, 74(1):6, 2024. [307] Mohammad Hamidi, Florentin Smarandache, and Elham Davneshvar. Spectrum of superhypergraphs via flows.Journal of Mathematics, 2022(1):9158912, 2022. [308] Takaaki Fujita and Florentin Smarandache.Neutrosophic soft n-super-hypergraphs with real-world appli- cations. Infinite Study, 2025. [309] Takaaki Fujita and Florentin Smarandache. Soft directed n-superhypergraphs with some real-world appli- cations.European Journal of Pure and Applied Mathematics, 18(4):6643–6643, 2025. [310] Ajoy Kanti Das, Rajat Das, Suman Das, Bijoy Krishna Debnath, Carlos Granados, Bimal Shil, and Rakhal Das. A comprehensive study of neutrosophic superhyper bci-semigroups and their algebraic significance. Transactions on Fuzzy Sets and Systems, 8(2):80, 2025. [311] Adel Al-Odhari. A brief comparative study on hyperstructure, super hyperstructure, and n-super super- hyperstructure.Neutrosophic Knowledge, 6:38–49, 2025. [312] Mohammad Hamidi and Mohadeseh Taghinezhad.Application of Superhypergraphs-Based Domination Number in Real World. Infinite Study, 2023. [313] Mohammad Hamidi, Florentin Smarandache, and Mohadeseh Taghinezhad.Decision Making Based on Valued Fuzzy Superhypergraphs. Infinite Study, 2023. [314] Takaaki Fujita, Atiqe Ur Rahman, Arkan A Ghaib, Talal Ali Al-Hawary, and Arif Mehmood Khattak. On the properties and illustrative examples of soft superhypergraphs and rough superhypergraphs.Prospects for Applied Mathematics and Data Analysis, 5(1):12–31, 2025. 140 Bibliography [315] Takaaki Fujita. Review of plithogenic directed, mixed, bidirected, and pangene offgraph.Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 120, 2024. [316] Takaaki Fujita. Recursive hypergraphs and recursive superhypergraphs: Exploring more hierarchical and generalized graph concepts. [317] Miguel Ortiz-Barrios, Natalia Jaramillo-Rueda, Andrea Espeleta-Aris, Berk Kucukaltan, and Llanos Cuenca. Integrated fuzzy decision-making methodology with intuitionistic fuzzy numbers: An applica- tion for disaster preparedness in clinical laboratories.Expert Systems with Applications, 263:125712, 2025. [318] Hongxing Li and Vincent C Yen.Fuzzy sets and fuzzy decision-making. CRC press, 1995. [319] Dragan Pamucar, Morteza Yazdani, Radojko Obradovic, Anil Kumar, and Mercedes Torres-Jiménez. A novel fuzzy hybrid neutrosophic decision-making approach for the resilient supplier selection problem. International Journal of Intelligent Systems, 35(12):1934–1986, 2020. [320] Arunodaya Raj Mishra, Dragan Pamucar, Pratibha Rani, Rajeev Shrivastava, and Ibrahim M. Hezam. Assessing the sustainable energy storage technologies using single-valued neutrosophic decision-making framework with divergence measure.Expert Syst. Appl., 238:121791, 2023. [321] G Muhiuddin, Mohamed E Elnair, Satham Hussain, and Durga Nagarajan. Topsis method-based decision- making model for bipolar quadripartitioned neutrosophic environment.Neutrosophic Sets and Systems, 85:899–918, 2025. [322] Muhammet Gul, Suleyman Mete, Faruk Serin, and Erkan Celik. Fine–kinney-based occupational risk as- sessment using single-valued neutrosophic topsis. InFine–Kinney-Based Fuzzy Multi-criteria Occupational Risk Assessment: Approaches, Case Studies and Python Applications, pages 111–133. Springer, 2020. [323] Ting-Yu Chen and Chueh-Yung Tsao. The interval-valued fuzzy topsis method and experimental analysis. Fuzzy sets and systems, 159(11):1410–1428, 2008. [324] Manoj Mathew, Ripon Kumar Chakrabortty, and Michael J. Ryan. A novel approach integrating ahp and topsis under spherical fuzzy sets for advanced manufacturing system selection.Eng. Appl. Artif. Intell., 96:103988, 2020. [325] Ali Azadeh, Morteza Saberi, Nasim Zandi Atashbar, Elizabeth Chang, and Peiman Pazhoheshfar. Z-ahp: A z-number extension of fuzzy analytical hierarchy process. In2013 7th IEEE International Conference on Digital Ecosystems and Technologies (DEST), pages 141–147. IEEE, 2013. [326] Hamid Reza Pourghasemi, Biswajeet Pradhan, and Candan Gokceoglu. Application of fuzzy logic and analytical hierarchy process (ahp) to landslide susceptibility mapping at haraz watershed, iran.Natural Hazards, 63:965–996, 2012. [327] Mavera Nawaz, Arooj Adeel, and Muhammad Akram. Risk evaluation in failure mode and effect analysis: Ahp-vikor method with picture fuzzy rough number.Granular Computing, 9(3):69, 2024. [328] Xingang Wang, Yushui Geng, Peipei Yao, and Mengjie Yang. Multiple attribute group decision making approach based on extended vikor and linguistic neutrosophic set.Journal of Intelligent & Fuzzy Systems, 36(1):149–160, 2019. [329] Serafim Opricovic and Gwo-Hshiung Tzeng. Compromise solution by mcdm methods: A comparative analysis of vikor and topsis.Eur. J. Oper. Res., 156:445–455, 2004. [330] Admin Admin, Luis A. Crespo Crespo-Berti, Haro Teran Lilian Fabiola, and Dinara Turaeva. Neutrosophic decision making using saaty’s ahp method and vikor.Journal of Intelligent Systems and Internet of Things, 2024. [331] Muhammad Riaz and Syeda Tayyba Tehrim. A robust extension of vikor method for bipolar fuzzy sets using connection numbers of spa theory based metric spaces.Artificial Intelligence Review, 54:561 – 591, 2020. 141 Soft set theory serves as a structured framework for parameterized decision modeling by associating specific attributes with subsets of a given universe, allowing for the effective representation of uncertainty. Over the past several decades, this theory has expanded significantly into various specialized models designed to handle increasingly complex data structures. This book presents a survey- style exploration of these developments, detailing core definitions and representative constructions for extensions such as: • HyperSoft and SuperHyperSoft Sets: Models that capture multi-attribute interactions and set-valued constraints. • TreeSoft and ForestSoft Sets: Hierarchical organizations that model refined parameters across multiple levels. • Dynamic Soft Sets: Time-indexed families of soft sets that model approximations as they evolve over time or context. • Uncertainty-Aware Models: Integrations with fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets. In addition to theoretical foundations, the book highlights key applications in diverse fields, including decision-making (such as AHP and TOPSIS), topology, matroid theory, and graph neural networks. It aims to organize the vast amount of existing research into a clear, accessible landscape for researchers and practitioners.