Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets

ISBN-13: 9789048163786 / Angielski / Miękka / 2010 / 470 str.

S.E. Rodabaugh;Erich Peter Klement

Opis

This volume summarizes recent developments in the topological and algebraic structures in fuzzy sets and may be rightly viewed as a continuation of the stan dardization of the mathematics of fuzzy sets established in the "Handbook," namely the Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Volume 3 of The Handbooks of Fuzzy Sets Series (Kluwer Academic Publish ers, 1999). Many of the topological chapters of the present work are not only based upon the foundations and notation for topology laid down in the Hand book, but also upon Handbook developments in convergence, uniform spaces, compactness, separation axioms, and canonical examples; and thus this work is, with respect to topology, a continuation of the standardization of the Hand book. At the same time, this work significantly complements the Handbook in regard to algebraic structures. Thus the present volume is an extension of the content and role of the Handbook as a reference work. On the other hand, this volume, even as the Handbook, is a culmination of mathematical developments motivated by the renowned International Sem inar on Fuzzy Set Theory, also known as the Linz Seminar, held annually in Linz, Austria. Much of the material of this volume is related to the Twenti eth Seminar held in February 1999, material for which the Seminar played a crucial and stimulating role, especially in providing feedback, connections, and the necessary screening of ideas.

Spis treści

Volume Contributors. Preface. Introduction. I: Topological Structures in Fuzzy Sets. 1. Uniform Completion in Pointfree Topology; B. Banaschewshi. 2. Monadic Convergence Structures; W. Gähler. 3. A Unified Approach to the Concept of Fuzzy L-Uniform Space; J. Gutiérrez Garcí, M.A. de Prada Vicente, A.P. Sostak. 4. Many Valued Topologies and Borel Probability Measures; U. Höhle. 5. Fuzzy Reals: Topological Results Surveyed, Brouwer Fixed Point Theorem, Open Questions; T. Kubiak. 6. Lattice-Valued Frames, Functor Categories and Classes of Sober Spaces; A. Pultr, S.E. Rodabaugh. Appendix to Chapter 6: Weakening the Requirement that L be a Complete Chain; U. Höhle, S.E. Rodabaugh. 7. Axiomatic Foundations for Uniform Operator Quasi-Uniformities; S.E. Rodabaugh. 8. Full Fuzzy Topology; L.N. Stout. II: Algebraic Structures in Fuzzy Sets. 9. Fuzzy Logics Arising from Strict De Morgan Systems; M. Gehrke, C. Walker, E. Walker. 10. Structure of Girard Monoids on [0,1]; S. Jenei. 11. On the Geometry of Choice; C.J. Mulvey. 12. On Some Fuzzy Categories Related to Category LTOP of L-Topological Spaces; A.P. Sostak. III: Related Topics in Topological and Algebraic Structures. 13. Fuzzy Compactness via Categorical Closure Operators; I.W. Alderton. 14. Discrete Triangular Norms; B. de Baets, R. Mesiar. 15. Powerset Operators Based Approach to Fuzzy Topologies on Fuzzy Sets; C. Guido. 16. Lifting of Sobriety Concepts with Particular Reference to (L,M)-Topological Spaces; W. Kotzé. 17. Examples for Different Sobrieties in Fixed-Basis Topology; A. Pultr, S.E. Rodabaugh. 18. Additive Generators of Non-Continuous Triangular Norms; P. Viceník. 19. Groups, T-Norms and Families of De Morgan Systems; C. Walker, E. Walker.