Triangular Norms (TRENDS IN LOGIC Volume 8)

The history of triangular norms started with the paper "Statistical metrics" [Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to de­ scribe the distance between two elements of the space in question. Triangular norms (t-norms for short) naturally came into the picture in the course of the generalization of the classical triangle inequality to this more general set­ ting. The original set of axioms for t-norms was considerably weaker, including among others also the functions which are known today as triangular conorms. Consequently, the first field where t-norms played a major role was the theory of probabilistic metric spaces ( as statistical metric spaces were called after 1964). Berthold Schweizer and Abe Sklar in [Schweizer & Sklar 1958, 1960, 1961] provided the axioms oft-norms, as they are used today, and a redefinition of statistical metric spaces given in [Serstnev 1962]led to a rapid development of the field. Many results concerning t-norms were obtained in the course of this development, most of which are summarized in the monograph [Schweizer & Sklar 1983]. Mathematically speaking, the theory of (continuous) t-norms has two rather independent roots, namely, the field of (specific) functional equations and the theory of (special topological) semigroups. The history of triangular norms started with the paper "Statistical metrics" [Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to deƯ scribe the distance between two elements of the space in question. Triangular norms (t-norms for short) naturally came into the picture in the course of the generalization of the classical triangle inequality to this more general setƯ ting. The original set of axioms for t-norms was considerably weaker, including among others also the functions which are known today as triangular conorms. Consequently, the first field where t-norms played a major role was the theory of probabilistic metric spaces (as statistical metric spaces were called after 1964). Berthold Schweizer and Abe Sklar in [Schweizer & Sklar 1958, 1960, 1961] provided the axioms oft-norms, as they are used today, and a redefinition of statistical metric spaces given in [Serstnev 1962]led to a rapid development of the field. Many results concerning t-norms were obtained in the course of this development, most of which are summarized in the monograph [Schweizer & Sklar 1983]. Mathematically speaking, the theory of (continuous) t-norms has two rather independent roots, namely, the field of (specific) functional equations and the theory of (special topological) semigroups Triangular norms were first used in the context of probabilistic metric spaces in order to extend the triangle inequality from classical metric spaces to this more general case. The theory of triangular norms has two roots, viz., specific functional equations and the theory of special topological semigroups. These are discussed in Part I. Part II of the book surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. Part I is self contained, including all proofs, and gives many graphical illustrations. The review in Part II shows the importance if triangular norms in the field concerned, providing a well-balanced picture of theory and applications. Front Matter....Pages i-xix Front Matter....Pages 1-1 Basic definitions and properties....Pages 3-19 Algebraic aspects....Pages 21-51 Construction of t-norms....Pages 53-100 Families of t-norms....Pages 101-119 Representations of t-norms....Pages 121-140 Comparison of t-norms....Pages 141-156 Values and discretization of t-norms....Pages 157-176 Convergence of t-norms....Pages 177-192 Front Matter....Pages 193-193 Distribution functions....Pages 195-214 Aggregation operators....Pages 215-228 Many-valued logics....Pages 229-247 Fuzzy set theory....Pages 249-264 Applications of fuzzy logic and fuzzy sets....Pages 265-282 Generalized measures and integrals....Pages 283-312 Back Matter....Pages 313-387 Acknowledgments Introduction Notations used in this book Part I: 1. Basic definitions and properties. 2. Algebraic aspects. 3. Construction of t-norms. 4. Families of t-norms. 5. Representations of t-norms. 6. Comparison of t-norms. 7. Values and discretization of t-norms. 8. Convergence of t-norms Part II: 9. Distribution functions. 10. Aggregation operators. 11. Many-valued logics. 12. Fuzzy set theory. 13. Applications of fuzzy logic and fuzzy sets. 14. Generalized measures and integrals Appendix A: Families of t-norms. B: Additional t-norms Reference material List of Figures List of Tables List of Mathematical Symbols Bibliography Index. This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.