Triangular Norms

by Klement, Erich Peter; Mesiar, Radko; Pap, Endre

• ISBN13: 9780792364160
• ISBN10: 0792364163
• Format: Hardcover
• Copyright: 2000-08-01
• Publisher: Kluwer Academic Pub

Summary

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.

Table of Contents

Acknowledgments	xi
Introduction	xiii
Notations used in this book	xvii
Part I
Basic definitions and properties	3	(18)
Triangular norms	4	(7)
Triangular conorms	11	(3)
Continuity	14	(7)
Algebraic aspects	21	(32)
Elementary algebraic properties	22	(13)
Semigroups and t-norms	35	(8)
Topological semigroups and continuous t-norms	43	(4)
Lattice-ordered monoids and left-continuous t-norms	47	(6)
Construction of t-norms	53	(48)
Pseudo-inverses of monotone functions	54	(14)
Additive and multiplicative generators	68	(13)
Ordinal sums	81	(11)
Constructions of non-continuous t-norms	92	(9)
Families of t-norms	101	(20)
Basic t-norms and t-conorms	102	(1)
Schweizer-Sklar t-norms	103	(2)
Hamacher t-norms	105	(3)
Frank t-norms	108	(2)
Yager t-norms	110	(2)
Dombi t-norms	112	(2)
Sugeno-Weber t-norms	114	(1)
Aczel-Alsina t-norms	115	(2)
Mayor-Torrens t-norms	117	(4)
Representations of t-norms	121	(20)
Representation of continuous Archimedean t-norms	122	(4)
Strict and nilpotent t-norms	126	(2)
Representation of continuous t-norms	128	(2)
Functional equations	130	(11)
Comparison of t-norms	141	(16)
Comparision of continuous Archimedean t-norms	142	(7)
Comparison of continuous t-norms	149	(3)
Domination of t-norms	152	(5)
Values and discretization of t-norms	157	(20)
Values and preimages	158	(5)
Sections and segments	163	(8)
Discrete t-norms	171	(6)
Convergence of t-norms	177	(18)
Approximation of continuous t-norms	178	(8)
Convergence of t-norms and their generators	186	(9)
Part II
Distribution functions	195	(20)
Copulas	197	(10)
Triangle functions	207	(3)
Probabilistic metric spaces	210	(5)
Aggregation operators	215	(14)
Triangular norm-based aggregation operators	216	(5)
Uninorms and nullnorms	221	(8)
Many-valued logics	229	(20)
Interpretations of connectives in fuzzy logics	230	(6)
Residuum-based fuzzy logics	236	(2)
Residuum-based fuzzy logics: Frank t-norms	238	(5)
S-fuzzy logics	243	(6)
Fuzzy set theory	249	(16)
Fuzzy subsets of a universe	250	(4)
T-equivalences and T-E-orderings	254	(5)
T-partitions	259	(2)
T-clans and T-tribes	261	(4)
Applications of fuzzy logic and fuzzy sets	265	(18)
Fuzzy relations, compositional rule of inference	266	(2)
Fuzzy numbers	268	(9)
Fuzzy control	277	(6)
Generalized measures and integrals	283	(72)
Measures on T-tribes	284	(7)
Decomposable measures	291	(5)
Integrals based on t-norms and t-conorms	296	(6)
Generalized convolution and Laplace transform	302	(5)
Information measures	307	(8)
Appendix
A. Families of t-norms	315	(18)
A.1 Aczel-Alsina t-norms	316	(2)
A.2 Dombi t-norms	318	(2)
A.3 Frank t-norms	320	(2)
A.4 Hamacher t-norms	322	(2)
A.5 Mayor-Torrens t-norms	324	(2)
A.6 Schweizer-Sklar t-norms	326	(2)
A.7 Sugeno-Weber t-norms	328	(2)
A.8 Yager t-norms	330	(3)
B. Additional t-norms	333	(12)
B.1 Krause t-norm	333	(7)
B.2 A family of incomparable t-norms	340	(5)
Reference material
List of Figures	345	(2)
List of Tables	347	(2)
List of Mathematical Symbols	349	(6)
Bibliography	355	(20)
Index	375