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

✍ Scribed by Stephen Ernest Rodabaugh, Erich Peter Klement (eds.)

• Publisher: Springer Netherlands
• Year: 2003
• Tongue: English
• Leaves: 467
• Series: Trends in Logic 20
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xi
Introduction....Pages 1-16
Front Matter....Pages 17-17
Uniform Completion In Pointfree Topology....Pages 19-56
Monadic Convergence Structures....Pages 57-79
A Unified Approach To The Concept Of Fuzzy L -Uniform Space....Pages 81-114
Many Valued Topologies And Borel Probability Measures....Pages 115-135
Fuzzy Reals: Topological Results Surveyed, Brouwer Fixed Point Theorem, Open Questions....Pages 137-151
Lattice-Valued Frames, Functor Categories, And Classes Of Sober Spaces....Pages 153-187
Weakening The Requirement That L Be A Complete Chain....Pages 189-197
Axiomatic Foundations For Uniform Operator Quasi-Uniformities....Pages 199-233
Fully Fuzzy Topology....Pages 235-253
Front Matter....Pages 255-255
Fuzzy Logics Arising From Strict De Morgan Systems....Pages 257-276
Structure Of Girard Monoids On [0,1]....Pages 277-308
On The Geometry Of Choice....Pages 309-336
Front Matter....Pages 373-373
On Some Fuzzy Categories Related To Category L -TOP Of L -Topological Spaces....Pages 337-371
Fuzzy Compactness Via Categorical Closure Operators....Pages 375-387
Discrete Triangular Norms....Pages 389-400
Powerset Operators Based Approach To Fuzzy Topologies On Fuzzy Sets....Pages 401-413
Lifting Of Sobriety Concepts With Particular Reference To ( L, M )-Topological Spaces....Pages 415-426
Examples For Different Sobrieties In Fixed-Basis Topology....Pages 427-440
Additive Generators Of Non-Continuous Triangular Norms....Pages 441-454
Front Matter....Pages 373-373
Groups, T-Norms, And Families Of De Morgan Systems....Pages 455-467
Back Matter....Pages 469-470

✦ Subjects

Mathematical Logic and Foundations; Geometry; Order, Lattices, Ordered Algebraic Structures; Group Theory and Generalizations; Logic