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Auteur Ulrich Höhle |
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Many Valued Topology and its Applications / Ulrich Höhle / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (2001)
Titre : Many Valued Topology and its Applications Type de document : document électronique Auteurs : Ulrich Höhle ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 2001 Importance : VII, 382 p Présentation : online resource ISBN/ISSN/EAN : 978-1-4615-1617-0 Langues : Anglais (eng) Tags : Mathematics Mathematical logic Topology Mathematical Logic and Foundations Résumé : The 20th Century brought the rise of General Topology. It arose from the effort to establish a solid base for Analysis and it is intimately related to the success of set theory. Many Valued Topology and Its Applications seeks to extend the field by taking the monadic axioms of general topology seriously and continuing the theory of topological spaces as topological space objects within an almost completely ordered monad in a given base category C. The richness of this theory is shown by the fundamental fact that the category of topological space objects in a complete and cocomplete (epi, extremal mono)-category C is topological over C in the sense of J. Adamek, H. Herrlich, and G.E. Strecker. Moreover, a careful, categorical study of the most important topological notions and concepts is given - e.g., density, closedness of extremal subobjects, Hausdorff's separation axiom, regularity, and compactness. An interpretation of these structures, not only by the ordinary filter monad, but also by many valued filter monads, underlines the richness of the explained theory and gives rise to new concrete concepts of topological spaces - so-called many valued topological spaces. Hence, many valued topological spaces play a significant role in various fields of mathematics - e.g., in the theory of locales, convergence spaces, stochastic processes, and smooth Borel probability measures. In its first part, the book develops the necessary categorical basis for general topology. In the second part, the previously given categorical concepts are applied to monadic settings determined by many valued filter monads. The third part comprises various applications of many valued topologies to probability theory and statistics as well as to non-classical model theory. These applications illustrate the significance of many valued topology for further research work in these important fields Note de contenu : I Categorical Foundations -- 1 Categorical Preliminaries -- 2 Partially Ordered Monads -- 3 Categorical Basis of Topology -- II Many Valued Topology -- 4 Quantic Basis of Filter Theory -- 5 Many Valued Topological Spaces -- 6 Many Valued Convergence Theory -- III Applications of Many Valued Topology -- 7 Stochastic Metrics -- 8 Stochastic Processes -- 9 Probability Measures -- 10 Topologies on M-Valued Sets -- A.1 Regularity based on ortholattices -- A.2 Topologization of Menger spaces -- Author Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=127467 Many Valued Topology and its Applications [document électronique] / Ulrich Höhle ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 2001 . - VII, 382 p : online resource.
ISBN : 978-1-4615-1617-0
Langues : Anglais (eng)
Tags : Mathematics Mathematical logic Topology Mathematical Logic and Foundations Résumé : The 20th Century brought the rise of General Topology. It arose from the effort to establish a solid base for Analysis and it is intimately related to the success of set theory. Many Valued Topology and Its Applications seeks to extend the field by taking the monadic axioms of general topology seriously and continuing the theory of topological spaces as topological space objects within an almost completely ordered monad in a given base category C. The richness of this theory is shown by the fundamental fact that the category of topological space objects in a complete and cocomplete (epi, extremal mono)-category C is topological over C in the sense of J. Adamek, H. Herrlich, and G.E. Strecker. Moreover, a careful, categorical study of the most important topological notions and concepts is given - e.g., density, closedness of extremal subobjects, Hausdorff's separation axiom, regularity, and compactness. An interpretation of these structures, not only by the ordinary filter monad, but also by many valued filter monads, underlines the richness of the explained theory and gives rise to new concrete concepts of topological spaces - so-called many valued topological spaces. Hence, many valued topological spaces play a significant role in various fields of mathematics - e.g., in the theory of locales, convergence spaces, stochastic processes, and smooth Borel probability measures. In its first part, the book develops the necessary categorical basis for general topology. In the second part, the previously given categorical concepts are applied to monadic settings determined by many valued filter monads. The third part comprises various applications of many valued topologies to probability theory and statistics as well as to non-classical model theory. These applications illustrate the significance of many valued topology for further research work in these important fields Note de contenu : I Categorical Foundations -- 1 Categorical Preliminaries -- 2 Partially Ordered Monads -- 3 Categorical Basis of Topology -- II Many Valued Topology -- 4 Quantic Basis of Filter Theory -- 5 Many Valued Topological Spaces -- 6 Many Valued Convergence Theory -- III Applications of Many Valued Topology -- 7 Stochastic Metrics -- 8 Stochastic Processes -- 9 Probability Measures -- 10 Topologies on M-Valued Sets -- A.1 Regularity based on ortholattices -- A.2 Topologization of Menger spaces -- Author Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=127467 Mathematics of Fuzzy Sets / Ulrich Höhle ; SpringerLink (Online service) ; Stephen Ernest Rodabaugh / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1999)
Titre : Mathematics of Fuzzy Sets : Logic, Topology, and Measure Theory Type de document : document électronique Auteurs : Ulrich Höhle ; SpringerLink (Online service) ; Stephen Ernest Rodabaugh Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1999 Collection : The Handbooks of Fuzzy Sets Series, ISSN 1388-4352 num. 3 Importance : XII, 716 p Présentation : online resource ISBN/ISSN/EAN : 978-1-4615-5079-2 Langues : Anglais (eng) Tags : Mathematics Operations research Decision making Mathematical logic Calculus of variations Mathematical Logic and Foundations Calculus of Variations and Optimal Control Optimization Operation Research Decision Theory Résumé : 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 Note de contenu : 1. Many-valued logic and fuzzy set theory -- 2. Powerset operator foundations for poslat fuzzy set theories and topologies -- Introductory notes to Chapter 3 -- 3. Axiomatic foundations of fixed-basis fuzzy topology -- 4. Categorical foundations of variable-basis fuzzy topology -- 5. Characterization of L-topologies by L-valued neighborhoods -- 6. Separation axioms: Extension of mappings and embedding of spaces -- 7. Separation axioms: Representation theorems, compactness, and compactifications -- 8. Uniform spaces -- 9. Extensions of uniform space notions -- 10. Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity -- 11. Fundamentals of generalized measure theory -- 12. On conditioning operators -- 13. Applications of decomposable measures -- 14. Fuzzy random variables revisited Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=127499 Mathematics of Fuzzy Sets : Logic, Topology, and Measure Theory [document électronique] / Ulrich Höhle ; SpringerLink (Online service) ; Stephen Ernest Rodabaugh . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1999 . - XII, 716 p : online resource. - (The Handbooks of Fuzzy Sets Series, ISSN 1388-4352; 3) .
ISBN : 978-1-4615-5079-2
Langues : Anglais (eng)
Tags : Mathematics Operations research Decision making Mathematical logic Calculus of variations Mathematical Logic and Foundations Calculus of Variations and Optimal Control Optimization Operation Research Decision Theory Résumé : 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 Note de contenu : 1. Many-valued logic and fuzzy set theory -- 2. Powerset operator foundations for poslat fuzzy set theories and topologies -- Introductory notes to Chapter 3 -- 3. Axiomatic foundations of fixed-basis fuzzy topology -- 4. Categorical foundations of variable-basis fuzzy topology -- 5. Characterization of L-topologies by L-valued neighborhoods -- 6. Separation axioms: Extension of mappings and embedding of spaces -- 7. Separation axioms: Representation theorems, compactness, and compactifications -- 8. Uniform spaces -- 9. Extensions of uniform space notions -- 10. Fuzzy real lines and dual real lines as poslat topological, uniform, and metric ordered semirings with unity -- 11. Fundamentals of generalized measure theory -- 12. On conditioning operators -- 13. Applications of decomposable measures -- 14. Fuzzy random variables revisited Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=127499 Semigroups in Complete Lattices / Patrik Eklund / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (2018)
Titre : Semigroups in Complete Lattices : Quantales, Modules and Related Topics Type de document : document électronique Auteurs : Patrik Eklund, ; Javier Gutiérrez García, ; SpringerLink (Online service) ; Ulrich Höhle, ; Jari Kortelainen, Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 2018 Collection : Developments in Mathematics, ISSN 1389-2177 num. 54 Importance : XXI, 326 p Présentation : online resource ISBN/ISSN/EAN : 978-3-319-78948-4 Langues : Anglais (eng) Tags : Algebra Computer science Order Lattices Ordered Algebraic Structures Category Theory Homological Algebra Mathematical Logic and Formal Languages Résumé : This monograph provides a modern introduction to the theory of quantales. First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research. This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic Note de contenu : Introduction -- 1 Foundations -- 2 Fundamentals of Quantales -- 3 Module Theory in Sup -- Appendix -- References -- Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=143666 Semigroups in Complete Lattices : Quantales, Modules and Related Topics [document électronique] / Patrik Eklund, ; Javier Gutiérrez García, ; SpringerLink (Online service) ; Ulrich Höhle, ; Jari Kortelainen, . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 2018 . - XXI, 326 p : online resource. - (Developments in Mathematics, ISSN 1389-2177; 54) .
ISBN : 978-3-319-78948-4
Langues : Anglais (eng)
Tags : Algebra Computer science Order Lattices Ordered Algebraic Structures Category Theory Homological Algebra Mathematical Logic and Formal Languages Résumé : This monograph provides a modern introduction to the theory of quantales. First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research. This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic Note de contenu : Introduction -- 1 Foundations -- 2 Fundamentals of Quantales -- 3 Module Theory in Sup -- Appendix -- References -- Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=143666