Topological And Algebraic Structures In Fuzzy Sets: A Handbook Of Recent Developments In The Mathematics Of Fuzzy Sets (trends In Logic)

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. 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 Topological And Algebraic Structures In Fuzzy Sets Has These Unique Features: -strategically Located At The Juncture Of Fuzzy Sets, Topology, Algebra, Lattices, Foundations Of Mathematics; -major Studies In Uniformities And Convergence Structures, Fundamental Examples In Lattice-valued Topology, Modifications And Extensions Of Sobriety, Categorical Aspects Of Lattice-valued Subsets, Logic And Foundations Of Mathematics, T-norms And Associated Algebraic And Ordered Structures; -internationally Recognized Authorities Clarify Deep Mathematical Aspects Of Fuzzy Sets, Particularly Those Topological Or Algebraic In Nature; -comprehensive Bibliographies And Tutorial Nature Of Longer Chapters Take Readers To The Frontier Of Each Topic; -extensively Referenced Introduction Unifies Volume And Guides Readers To Chapters Closest To Their Interests; -annotated Open Questions Direct Future Research In The Mathematics Of Fuzzy Sets; -suitable As A Text For Advanced Graduate Students. Edited By Stephen Ernest Rodabaugh, Erich Peter Klement.