Associative functions : triangular norms and copulas

The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the "triangular norms" introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis. This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations. The dynamics of complex systems can clarify the creation of structures in Nature. This creation is driven by the collective interaction of constitutive elements of the system. Such interactions are frequently nonlinear and are directly responsible for the lack of prediction in the evolution process. The self-organization accompanying these processes occurs all around us and is constantly being rediscovered, under the guise of a new jargon, in apparently unrelated disciplines. This volume offers unique perspectives on aspects of fractals and complexity and, through the examination of complementary techniques, provides a unifying thread in this multidisciplinary endeavour. Do nonlinear interactions play a role in the complexity management of socio-economic-political systems? Is it possible to extract the global properties of genetic regulatory networks without knowing the details of individual genes? What can one learn by transplanting the self-organization effects known in laser processes to the study of emotions? What can the change in the level of complexity tell us about the physiological state of the organism? The reader will enjoy finding the answers to these questions and many more in this book Preface; Special Symbols; Contents; 1. Introduction; 1.1 Historical notes; 1.2 Preliminaries; 1.3 t-norms and s-norms; 1.4 Copulas; 2. Representation theorems for associative functions; 2.1 Continuous Archimedean t-norms; 2.2 Additive and multiplicative generators; 2.3 Extension to arbitrary closed intervals; 2.4 Continuous non-Archimedean t-norms; 2.5 Non-continuous t-norms; 2.6 Families of t-norms; 2.7 Other representation theorems; 2.8 Related functional equations; 3. Functional equations involving t-norms; 3.1 Simultaneous associativity; 3.2 n-duality; 3.3 Simple characterizations of Min "This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations."--BOOK JACKET 3.4 Homogeneity3.5 Distributivity; 3.6 Conical t-norms; 3.7 Rational Archimedean t-norms; 3.8 Extension and sets of uniqueness; 4. Inequalities involving t-norms; 4.1 Notions of concavity and convexity; 4.2 The dominance relation; 4.3 Uniformly close associative functions; 4.4 Serial iterates and n-copulas; 4.5 Positivity; Appendix A Examples and counterexamples; Appendix B Open problems; Bibliography; Index