Triangular Norms

  • Erich Peter Klement
  • Radko Mesiar
  • Endre Pap

Part of the Trends in Logic book series (TREN, volume 8)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Part I

    1. Front Matter
      Pages 1-1
    2. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 3-19
    3. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 21-51
    4. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 53-100
    5. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 101-119
    6. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 121-140
    7. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 141-156
    8. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 157-176
    9. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 177-192
  3. Part II

    1. Front Matter
      Pages 193-193
    2. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 195-214
    3. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 215-228
    4. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 229-247
    5. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 249-264
    6. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 265-282
    7. Erich Peter Klement, Radko Mesiar, Endre Pap
      Pages 283-312
  4. Back Matter
    Pages 313-387

About this book

Introduction

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.

Keywords

addition algebra logic proof semigroup set theory

Authors and affiliations

  • Erich Peter Klement
    • 1
  • Radko Mesiar
    • 2
    • 4
  • Endre Pap
    • 3
  1. 1.Johannes Kepler UniversityLinzAustria
  2. 2.Slovak University of TechnologyBratislavaSlovakia
  3. 3.University of Novi SadYugoslavia
  4. 4.Czech Academy of SciencesPragueCzech Republic

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-9540-7
  • Copyright Information Springer Science+Business Media B.V. 2000
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5507-1
  • Online ISBN 978-94-015-9540-7
  • Series Print ISSN 1572-6126
  • Series Online ISSN 2212-7313
  • About this book