Triangular Norms

Erich Peter Klement, R. Mesiar, E. Pap (Autoren)

Buch | Softcover

387 Seiten

2010 | Softcover reprint of the original 1st ed. 2000
Springer (Verlag)
978-90-481-5507-1 (ISBN)

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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.

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.

Erscheint lt. Verlag	7.12.2010
Reihe/Serie	Trends in Logic ; 8
Zusatzinfo	XIX, 387 p.
Verlagsort	Dordrecht
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Geisteswissenschaften ► Philosophie ► Allgemeines / Lexika
	Geisteswissenschaften ► Philosophie ► Logik
	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
ISBN-10	90-481-5507-X / 904815507X
ISBN-13	978-90-481-5507-1 / 9789048155071
Zustand	Neuware