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＜E-Book＞
Triangular Norms / by Erich Peter Klement, R. Mesiar, E. Pap
(Trends in Logic, Studia Logica Library. ISSN:22127313 ; 8)

Edition	1st ed. 2000.
Publisher	(Dordrecht : Springer Netherlands : Imprint: Springer)
Year	2000
Language	English
Size	XIX, 387 p : online resource
Authors	*Klement, Erich Peter author Mesiar, R author Pap, E author SpringerLink (Online service)
Subjects	LCSH:Logic LCSH:Algebra LCSH:Mathematical logic FREE:Logic FREE:Order, Lattices, Ordered Algebraic Structures FREE:Mathematical Logic and Foundations
Notes	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 -- 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 -- A. Families of t-norms -- A.1 Aczél-Alsina t-norms -- A.2 Dombi t-norms -- A.3 Frank t-norms -- A.4 Hamacher t-norms -- A.5 Mayor-Torrens t-norms -- A.6 Schweizer-Sklar t-norms -- A.7 Sugeno-Weber t-norms -- A.8 Yager t-norms -- B. Additional t-norms -- B.1 Krause t-norm -- B.2 A family of incomparable t-norms -- Reference material -- List of Figures -- List of Tables -- List of Mathematical Symbols 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 HTTP:URL=https://doi.org/10.1007/978-94-015-9540-7

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