| Triangular Norms 2000 Edition Contributor(s): Klement, Erich Peter (Author), Mesiar, R. (Author), Pap, E. (Author) |
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ISBN: 0792364163 ISBN-13: 9780792364160 Publisher: Springer OUR PRICE: $208.99 Product Type: Hardcover - Other Formats Published: July 2000 Annotation: 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Calculus |
| Dewey: 515.724 |
| LCCN: 00033077 |
| Series: Trends in Logic |
| Physical Information: 0.94" H x 6.14" W x 9.21" (1.66 lbs) 387 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |