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Trends in Logic

Triangular Norms

Authors: Klement, Erich Peter, Mesiar, Radko, Pap, Endre

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About this book

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.

Reviews

"The book is very well written and constitutes a valuable addition to the literature on t-norms and, in general fuzzy reasoning. It presents almost all of the most important developments of fuzzy sets and fuzzy logic by focusing on the central concept of triangular norm. It is divided into two parts and an appendix, which gives a summary of the existing families of t-norms."
(Mathematical Reviews, 2002a)

Table of contents (14 chapters)

  • Basic definitions and properties

    Klement, Erich Peter (et al.)

    Pages 3-19

  • Algebraic aspects

    Klement, Erich Peter (et al.)

    Pages 21-51

  • Construction of t-norms

    Klement, Erich Peter (et al.)

    Pages 53-100

  • Families of t-norms

    Klement, Erich Peter (et al.)

    Pages 101-119

  • Representations of t-norms

    Klement, Erich Peter (et al.)

    Pages 121-140

Buy this book

eBook $269.00
price for USA (gross)
  • ISBN 978-94-015-9540-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $349.00
price for USA
  • ISBN 978-0-7923-6416-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $349.00
price for USA
  • ISBN 978-90-481-5507-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Triangular Norms
Authors
Series Title
Trends in Logic
Series Volume
8
Copyright
2000
Publisher
Springer Netherlands
Copyright Holder
Springer Science+Business Media Dordrecht
eBook ISBN
978-94-015-9540-7
DOI
10.1007/978-94-015-9540-7
Hardcover ISBN
978-0-7923-6416-0
Softcover ISBN
978-90-481-5507-1
Series ISSN
1572-6126
Edition Number
1
Number of Pages
XIX, 387
Topics