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Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms

1st Edition - March 25, 2005

Editors: Erich Peter Klement, Radko Mesiar

Hardback ISBN:

9 7 8 - 0 - 4 4 4 - 5 1 8 1 4 - 9

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 4 5 9 5 3 - 0

This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16… Read more

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This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces.Key Features:- Complete state of the art of the importance of triangular norms in various mathematical fields- 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications- Chapter authors are leading authorities in their fields- Triangular norms on different domains (including discrete, partially ordered) are described- Not only triangular norms but also related operators (aggregation operators, copulas) are covered- Book contains many enlightening illustrations

Preface
Part I: Introduction
- 1: Triangular norms, looking back—triangle functions, looking ahead
  - 1.1
  - 1.2
  - 1.3
  - 1.4
  - 1.5
  - 1.6
  - 1.7
  - 1.8
- 2: Triangular norms: Basic notions and properties
  - Abstract
  - 2.1 Introduction
  - 2.2 Triangular norms
  - 2.3 Triangular conorms
  - 2.4 Continuity
  - 2.5 Elementary algebraic properties
  - 2.6 Additive and multiplicative generators
  - 2.7 Ordinal sums
  - 2.8 Other constructions
  - 2.9 Families of t-norms and t-conorms
  - 2.10 Representation of continuous Archimedean t-norms
  - 2.11 Representation of continuous t-norms
  - 2.12 Consequences of the representation theorems
  - Acknowledgment
Part II: Theoretical Aspects of Triangular Norms
- 3: Semigroups and triangular norms
  - Abstract
  - 3.1 Introduction
  - 3.2 Semigroups
  - 3.3 Triangular norms as semigroup operations
  - 3.4 Subsemigroups and Archimedean components
  - 3.5 Ordinal sums
  - 3.6 I-semigroups
  - 3.7 Transformation of t-norms
  - Acknowledgment
- 4: Generators of triangular norms
  - Abstract
  - 4.1 Introduction and historical overview
  - 4.2 Additive generators of t-norms
  - 4.3 Representation of t-norms by means of additive generators
  - 4.4 Some characterizations of additive generators
  - 4.5 Weak additive generators of t-norms
  - 4.6 Conclusion and further perspectives
  - Acknowledgement
- 5: A survey on left-continuous t-norms and pseudo t-norms
  - Abstract
  - 5.1 Introduction
  - 5.2 Preliminaries
  - 5.3 Ordinal sums of t-subnorms
  - 5.4 Annihilation
  - 5.5 Rotation
  - 5.6 Rotation-annihilation
  - 5.7 Embedding method
  - 5.8 On the determination of left-continuous t-norms on some segments
  - Acknowledgment
- 6: Some aspects of functional equations
  - Abstract
  - 6.1 Introduction
  - 6.2 Reduction to differential equations
  - 6.3 Regularity properties
  - 6.4 Regularity and associativity
  - 6.5 Restricted domains
  - 6.6 Uniqueness results
  - 6.7 Copulas
  - 6.8 Fuzzy connectives
  - 6.9 Bisymmetry
  - 6.10 Concluding remarks
- 7: Triangular norms on discrete settings
  - Abstract
  - 7.1 Introduction
  - 7.2 Triangular norms on bounded chains
  - 7.3 Finitely-valued t-norms
  - 7.4 Countably infinite-valued t-norms
  - Acknowledgment
- 8: Triangular norms and related operators in L*-fuzzy set theory
  - Abstract
  - 8.1 Introduction
  - 8.2 The lattice L*
  - 8.3 Operators on L*
  - 8.4 The residuation principle for t-norms on L*
  - 8.5 Properties of implicators on L*
  - 8.6 Representable uninorms on L*
  - 8.7 Ordinal sums of t-norms on L*
  - 8.8 Conclusion
- 9: Fitting triangular norms to empirical data
  - Abstract
  - 9.1 Introduction
  - 9.2 Formulation of the regression problem
  - 9.3 Fitting continuous t-norms
  - 9.4 Approximation of additive generators
  - 9.5 Uninorms and Nullnorms
  - 9.6 Unknown neutral element/annihilator
  - 9.7 Conclusion
Part III: Applications of Triangular Norms and Related Operations
- 10: Triangular norm-based mathematical fuzzy logics
  - Abstract
  - 10.1 Introduction
  - 10.2 T-norm-based connectives
  - 10.3 Extracting an algebraic framework
  - 10.4 The logic of continuous t-norms
  - 10.5 The logic of left-continuous t-norms
  - 10.6 Particular cases of t-norm-based systems
  - 10.7 Computational complexity and other topics
  - 10.8 Expanding the language
  - 10.9 Further generalizations
  - 10.6 Allowing graded notions of consequence
  - 10.7 Conclusion
  - Acknowledgement
- 11: Many-valued equalities and their representations
  - Abstract
  - 11.1 Introduction
  - 11.2 Lattice-theoretic preliminaries
  - 11.3 Many-valued equalities and global extent of existence
  - 11.4 M-Valued sets and their representations
- 12: Varieties of algebras in fuzzy set theory
  - Abstract
  - 12.1 Introduction
  - 12.2 Preliminaries
  - 12.3 The basic lattices
  - 12.4 Lattices with a negation
  - 12.5 The unit interval with a t-norm
  - 12.6 Distributive lattice ordered commutative monoids
  - 12.7 Equations in two variables
  - 12.5 Varieties generated by De Morgan systems
  - Added in proof:
- 13: Triangular norms and measures of fuzzy sets
  - Abstract
  - 13.1 Introduction
  - 13.2 Tribes
  - 13.3 Charges on tribes
  - 13.4 Constructions with tribes and charges
  - 13.5 Preliminary results, Frank and nearly Frank t-norms
  - 13.6 σ-order continuous charges on full tribes
  - 13.7 Charges on full tribes
  - 13.8 σ-order continuous charges on general tribes
  - 13.9 Charges on general tribes
  - 3.10 Related topics, historical and bibliographical remarks
  - Acknowledgement
- 14: Copulas and quasi-copulas: An introduction to their properties and applications
  - Abstract
  - 14.1 Introduction
  - 14.2 Copulas
  - 14.3 Families of copulas
  - 14.4 Archimedean copulas
  - 14.5 Shuffles of M
  - 14.6 Descriptions of dependence and measures of association
  - 14.7 The Fréchet-Hoeffding bounds revisited
  - 14.8 Quasi-copulas
  - 14.9 Multivariate copulas and quasi-copulas
  - 14.10 Some open problems
- 15: Transitive comparison of random variables
  - Abstract
  - 15.1 Introduction
  - 15.2 Transitivity of probabilistic relations
  - 15.3 Discrete and continuous dice models
  - 15.4 Comparison of special random variables
  - 15.5 Dependent random variables
  - 15.6 Transitive comparison of dependent random variables
  - 15.7 Conclusion
  - Acknowledgment
- 16: Triangular norms in probabilistic metric spaces and fixed point theory
  - Abstract
  - 16.1 Introduction
  - 16.2 Probabilistic metric spaces
  - 16.3 Probabilistic q-contraction type mappings
  - 16.4 Triangular norms of H-type
  - 16.5 Geometrically convergent triangular norms
  - Acknowledgement
Index