Mathematics of Fuzzy Sets : Logic, Topology, and Measure Theory

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CONTENTS Authors and Editors Foreword Series Foreword Introduction 1. Many-Valued Logic And Fuzzy Set Theory, S. GOTTWALD 1. Basic ideas 1.1 From classical to many-valued logic 1.2 Truth degrees 1.3 Designated truth degrees 1.4 Logical validity and logical consequence 2. Outline of the history 3. Characteristic connectives 3.1 The J-connectives 3.2 Conjunction connectives 3.3 Negation connectives 3.4 Disjunction connectives 3.5 Implication connectives 4. Many-valued propositional logics 4.1 The ŁUKASIEWICZ systems 4.2 The GÖDEL systems 4.3 The product logic 4.4 The POST systems 4.5 A partial generalization: monoidal logic 4.6 A uniform method of axiomatisation 5. Algebraic structures for many-valued logics 5.1 MV-algebras 5.2 Łukasiewicz algebras 5.3 Product algebras 5.4 POST algebras 5.5 DEMoRGAN algebras 5.6 Residuated ℓ-monoids 6. Many-valued first-order logic 6.1 Basic notions 6.2 ŁUKASIEWICZ'S first-order logics 6.3 First-order monoidal logic 6.4 The uniform axiomatisation method of ROSSER/TURQUETTE 6.5 Adding identity 7. Fuzzy Sets and Many-Valued Logic 7.1 Membership Degrees as Truth Degrees 7.2 Doing Fuzzy Set Theory with MVL Language 8. Fuzzy logic 8.1 Adding partially sound inference rules to many-valued logic 8.2 Formalising the problem 8.3 Partially sound rules in many-valued logic 8.4 Many-valued logic with graded consequences References [6] [21] [38] [54] [73] [91] [107] 2. Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies, S. E. RODABAUGH Introduction 1. History, motivation, and preliminaries 2. Fuzzy powerset operators for fixed-basis poslat set theories 3. Fuzzy powerset operators for variable-basis poslat set theories 4. Powerset operator characterizations of ground isomorphisms 5 Fuzzy powerset operators and fuzzy associative memories References [17] Introductory Notes To Chapter 3, U. HÖHLE References [9] 3. Axiomatic Foundations of Fixed-Basis Fuzzy Topology, U. HÖHLE AND A.P. ŠOSTAK Introduction 1. Lattice-theoretic foundations. 1.1 Quantales. 1.2 Enriched cqm-lattices. 1.3 Complete MV-algebras with square roots and idempotent hulls. 1.4 Booleanization of bounded distributive lattices. 2. L-fuzzy topological spaces. 3. L-Topological spaces 4. Coreflective subcategories of L-FTOP. 4.1 Extensional L-fuzzy topologies. 4.2 Enriched, L-fuzzy topologies. 4.3 Strongly enriched L-fuzzy topologies 5. Coreflective subcategories of L-TOP 5.1 Stratified L-topologies 5.2 Strongly stratified L-topologies 6. Convergence theory for L-topological spaces and its applications 6.1 L-Interior operators and L- neighborhood systems 6.2 L-Filter theory 6.3 The principle of L-continuous extension 6.4 Compactness and stratified L-topological spaces 6.5 A level-wise characterization of L-neighborhood axioms 7. Examples of L-topological spaces 7.1 Case of complete Heyting algebras 7.2 Case of complete MV-algebras 7.3 Case of complete MV-algebras with square roots and rigid L-topologies 7.4 Lower semicontinuous, lattice-valued maps 8. Convergence theory for L-fuzzy topological spaces 8.1 L-fuzzy interior operators and L-fuzzy neighborhood systems 8.2 Convergence in L-fuzzy topological spaces 9 Local convergence theory for weakly exstensional 9.1 Extensionality and singletons in L^X 9.2 A representation theory for weakly extensional L-fuzzy topologies 9.3 Local L-interior operators 9.4 Filter theory on local elements of L^X 9.5 Hausdorff separation axiom for weakly extensional, L-fuzzy topological spaces 10. Historical comments 10.1 L-topologies, stratified L-topologies 10.2 L-interior operators, L-neighborhood structure 10.3 Convergence in L-topologies 10.4 Topological properties of L-topological spaces 10.5 Probabilistic L-topologies 10.6 [0,1]-topologies on spaces of probability measures 10.7 L-fuzzy topological spaces 10.8 The use of "enriched" lattices 10.9 Role of algebraic properties of the meet operator in the theory of L-topologies 10.10 Categorical aspects of Fuzzy Topology. Functors between categories of L-(fuzzy) topological spaces References [3] [19] [36] [51] [68] [85] [102) [117] 4. Categorical Foundations Of Variable-Basis Fuzzy Topology, S. E. RODABAUGH Introduction 1. Preliminary discussion and motivation 1.1 History of variable-basis thinking 1.2 Categorical motivation for variable-basis theories 1.3 Mathematical preliminaries and foundations 2. Ground categories SET x C and SET x L_Φ 2.1 Definitions of SET x C and SET x L_Φ 2.2 Categorical properties of SET x C and SET x L_Φ 3. Topological categories for variable-basis topology and fuzzy topology 3.1 Definitions of C-TOP and C-FTOP with examples of objects 3.2 Subbasic continuities and final structures in C-TOP and C-FTOP 3.3 C-TOP and C-FTOP are topological over SET x C 3.4 Categorical consequences of topological and fibresmallness 4. Topological categories for internalized change-of-basis 4.1 L_Φ-TOP and endomorphism-saturated spaces 4.2 L_Φ-FTOP and endomorphism-saturated fuzzy topologies 4.3 L_Φ-TOP and L_Φ-FTOP are topological over SET x L_Φ 4.4 Categorical consequences of topological and fibre-smallness 5. Categorical isomorphisms and embeddings 5.1 Homeomorphisms, fuzzy homeomorphisms, and categorical isomorphisms 5.2 Subspaces, initial structures, and initial morphisms 5.3 Topological, fuzzy topological, and categorical embeddings 6. Unification of topology and fuzzy topology by C-TOP and C-FTOP 6.1 Roster of subcategories 6.2 Functorial embeddings onto subcategories 6.3 Topological subcategories 7. Unification of canonical examples by C-TOP and C-FTOP 7.1 Fuzzy real lines and unit intervals 7.2 Generated (weakly stratified) poslat spaces 7.3 (Co-fuzzy) Dual real lines and unit intervals 7.4 Soberifications of spaces with different underlying bases 8. Acknowledgements References [14] [32] [50] [64] [79] 5. Characterization Of L-Topologies By L-Valued Neighborhoods, U. HÖHLE Introduction 1. Lattice-theoretic prerequisites 2. L-Filters in the case of (L, ≤, ∧, *) 3. Stratified and transitive L-topologies 4. Categorical properties of stratified and transitive L-topological spaces 5. Case of the real unit interval 6. Concluding Remark References [10] [26] 6. Separation Axioms: Extension Of Mappings And Embedding Of Spaces, T. KUBIAK Introduction 1. First L-topological concepts 2. L-real-valued functions 3. Separation axioms 4. Insertion and extension of mappings 5. Embedding of L-topological spaces References [10] [27] [44] 7. Separation Axioms: Representation Theorems, Compactness, And Compactifications, S. E. RODABAUGH Introduction 1. Views and approaches to compactification 2. Fixed-basis sobriety and spatiality 3. Separation axioms for representation and compactification 4. Compactness axioms for representation and compactification 5. Fixed-basis Stone representation theorems fordistributive lattices 6. Fixed-basis space representations of compact regular locales and compact Hausdorff spaces 7. Fixed-basis Stone representation theorems for Boolean algebras 8. Compactification reflectors for entire fixed basis categories of topology 9. Compactification reflectors for variable-basis categories of topology References [10] [28] [43] 8. Uniform Spaces, W. KOTZÉ Introduction 1. Uniform spaces 1.1 Diagonal uniformities 1.2 The uniform topology 2. Equivalent approaches to uniformity 2.1 Families of functions 2.2 Coverings 2.3 Uniformly continuous functions 3. Hutton L-fuzzy uniformities 3.1 Basic notions 3.2 Further results in the case of complete distributivity of the underlying lattice 4. Lowen fuzzy uniform spaces 4.1 I-Fuzzy uniform spaces 4.2 Fuzzy neighbourhood spaces 4.3 Fuzzy uniform topology 4.4 Uniformly continuous functions 5. L-Uniform spaces References [18] 9. Extensions Of Uniform Space Notions, M. H. BURTON AND J. GUTIÉRREZ GARCÍA Introduction 1. Preliminaries 2. I-Fuzzy uniform spaces 3. Cauchy filters 4. Precompactness 5. Boundedness 6. Completeness References [3] [20] [37] 10. Fuzzy Real Lines And Dual Real Lines As Poslat Topological, Uniform, And Metric Ordered Semirings With Unity, S. E. RODABAUGH Introduction 1. Preliminary notions of ℝ(L) and ℝ as poslat spaces 2. Basic tools for fuzzy arithmetic operations 3. ℝ(L) as L-topological and L-uniform additive monoid 4. ℝ(L) as L-topological complete fuzzy hyperfield 5. ℝ as L-uniform additive group and L-topological field References [12] [29] 11. Fundamentals of a Generalized Measure Theory, E. P. KLEMENT AND S. WEBER Introduction 1. Uncertainty measures and integration 2. Plausibility measures 3. Valuations 4. Additive measures on MV-algebras 5. Measures on tribes of fuzzy sets Concluding remarks References [3] [20] [36] 12. On Conditioning Operators, U. HÖRLE AND S. WEBER Introduction 1. Lattices of events 2. Interval based conditional events in the case of MV-algebras 3. Canonical extension of Girard algebras 4. Conditioning and mean value generation 5. Measure-free conditioning on canonical extensions of Girard algebras 6. Uncertainty measures and measure-free conditioning References [2] 13. Applications Of Decomposable Measures, E. PAP Introduction 1. Non-additive measures and integrals 1.1 Null-additive measures 1.2 Pseudo operations 1.3 Pseudo-integral 2. Pseudo-convolution and pseudo-Laplace transform 2.1 Pseudo-convolution 2.2 Pseudo-Laplace transform 3. Optimization 4. Morphism between probability calculus and decision calculus 5. Hamilton-Jacobi equation with non-smooth Hamiltonian 5.1 Two examples 5.2 General case References [16] [33] 14. Fuzzy Random Variables Revisited, D. A. RALESCU Introduction 1. The law of large numbers 2. Critical discussion of other results 3. The Brunn-Minkowski and the Jensen inequalities 4 Conclusions References [12] [30] Index ABC DEFGH IJKL MNOP QRSTU VWZ