Mathematics of Fuzzy Sets

Logic, Topology, and Measure Theory

  • Ulrich Höhle
  • Stephen Ernest Rodabaugh

Part of the The Handbooks of Fuzzy Sets Series book series (FSHS, volume 3)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Ulrich Höhle, Stephen Ernest Rodabaugh
    Pages 1-3
  3. U. Höhle
    Pages 117-122
  4. U. Höhle, A. P. Šostak
    Pages 123-272
  5. W. Kotzé
    Pages 553-580
  6. M. H. Burton, J. Gutiérrez García
    Pages 581-606
  7. E. P. Klement, S. Weber
    Pages 633-651
  8. U. Höhle, S. Weber
    Pages 653-673
  9. D. A. Ralescu
    Pages 701-710
  10. Back Matter
    Pages 711-716

About this book

Introduction

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14).
Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications.
Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact or complete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval.
Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton&endash;Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.

Keywords

Compactification Separation axiom fuzzy sets mathematical logic mathematics optimization probability theory set theory sets topology

Editors and affiliations

  • Ulrich Höhle
    • 1
  • Stephen Ernest Rodabaugh
    • 2
  1. 1.Fachbereich MathematikBergische UniversitätWuppertalGermany
  2. 2.Department of Mathematics and StatisticsYoungstown State UniversityYoungstownUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-5079-2
  • Copyright Information Kluwer Academic Publishers 1999
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-7310-0
  • Online ISBN 978-1-4615-5079-2
  • Series Print ISSN 1388-4352
  • About this book