Applications of Category Theory to Fuzzy Subsets: 14 (Theory and Decision Library B, 14) - Hardcover

Hardcover

ISBN 10: 0792315111 ISBN 13: 9780792315117

Publisher: Springer, 1991

Synopsis

This book has a fundamental relationship to the International Seminar on Fuzzy Set Theory held each September in Linz, Austria. First, this volume is an extended account of the eleventh Seminar of 1989. Second, and more importantly, it is the culmination of the tradition of the preceding ten Seminars. The purpose of the Linz Seminar, since its inception, was and is to foster the development of the mathematical aspects of fuzzy sets. In the earlier years, this was accomplished by bringing together for a week small grou ps of mathematicians in various fields in an intimate, focused environment which promoted much informal, critical discussion in addition to formal presentations. Beginning with the tenth Seminar, the intimate setting was retained, but each Seminar narrowed in theme; and participation was broadened to include both younger scholars within, and established mathematicians outside, the mathematical mainstream of fuzzy sets theory. Most of the material of this book was developed over the years in close association with the Seminar or influenced by what transpired at Linz. For much of the content, it played a crucial role in either stimulating this material or in providing feedback and the necessary screening of ideas. Thus we may fairly say that the book, and the eleventh Seminar to which it is directly related, are in many respects a culmination of the previous Seminars.

About this title

Synopsis

"Applications of Category Theory to Fuzzy Subsets" is the first major work to comprehensively describe the deeper mathematical aspects of fuzzy sets, particularly those aspects which are category-theoretic in nature, and is intimately related to the first eleven years of the renowned International Seminar on Fuzzy Set Theory. Though it brings the reader to the very frontier of the mathematics of fuzzy set theory, its extensive bibliography, indices, and the tutorial nature of its longer chapters also make it suitable as a text for advanced graduate students. Part I develops model-theoretic foundations for fuzzy set theory, and in doing so, comprises an extensive study of monoid-valued sets, sheaves over commutative cl-monoids, weak and quasi topoi, local existence in such settings, and categories with two closed structures, including the logic and inference rules in these latter categories for the unbalanced subobjects modeling fuzzy subsets.

Part II refines and works within non-model-theoretic approaches to fuzzy sets, giving a full account of the use of categorical methods to describe fuzzy topology from the structure-theoretic, category-theoretic, and point-set lattice-theoretic viewpoints. Explored in detail are set functors, topological constructs, convergence, and the relationship between locales, fuzzy topologies, and functor categories. Part III addresses issues related to Part I and II, including pointless geometry, fuzzy paths and relational databases, fuzzy points and locales, sobriety and Urysohn Lemmas, and the relationship between the Helly space and the fuzzy unit interval. The fourth part consists of several appendices: round tables and open questions, a comprehensive bibliography, and two extensive indices.

Edition details

• Publisher: Springer
• Publication date: 1991
• ISBN 10: 0792315111
• ISBN 13: 9780792315117
• Binding: Hardcover
• Language: English
• Number of pages: 407
• Editor: Rodabaugh S.E., Klement Erich Peter, Höhle Ulrich