Associative Functions: Triangular Norms and Copulas

Claudi Alsina, Maurice J Frank, Berthold Schweizer9789812566713, 981-256-671-6

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.

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Table of contents :
Contents……Page 14
Preface……Page 8
Special Symbols……Page 12
1.1 Historical notes……Page 16
1.2 Preliminaries……Page 21
1.3 t-norms and s-norms……Page 24
1.4 Copulas……Page 32
2.1 Continuous Archimedean t-norms……Page 38
2.2 Additive and multiplicative generators……Page 53
2.3 Extension to arbitrary closed intervals……Page 66
2.4 Continuous non-Archimedean t-norms……Page 72
2.5 Non-continuous t-norms……Page 79
2.6 Families of t-norms……Page 85
2.7 Other representation theorems……Page 96
2.8 Related functional equations……Page 108
3.1 Simultaneous associativity……Page 114
3.2 n-duality……Page 125
3.3 Simple characterizations of Min……Page 142
3.4 Homogeneity……Page 144
3.5 Distributivity……Page 149
3.6 Conical t-norms……Page 152
3.7 Rational Archimedean t-norms……Page 158
3.8 Extension and sets of uniqueness……Page 166
4.1 Notions of concavity and convexity……Page 188
4.2 The dominance relation……Page 197
4.3 Uniformly close associative functions……Page 204
4.4 Serial iterates and n-copulas……Page 209
4.5 Positivity……Page 218
Appendix A Examples and counterexamples……Page 224
Appendix B Open problems……Page 234
Bibliography……Page 238
Index……Page 250