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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.
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 Jacobi equation and optimization
problems. Chapter 14 surveys briefly the fundamentals of fuzzy
random variables which are [0,1]-valued interpretations of random
sets.
Non-Classical Logics and their Applications to Fuzzy Subsets is the
first major work devoted to a careful study of various relations
between non-classical logics and fuzzy sets. This volume is
indispensable for all those who are interested in a deeper
understanding of the mathematical foundations of fuzzy set theory,
particularly in intuitionistic logic, Lukasiewicz logic, monoidal
logic, fuzzy logic and topos-like categories. The tutorial nature
of the longer chapters, the comprehensive bibliography and index
make it suitable as a valuable and important reference for graduate
students as well as research workers in the field of non-classical
logics. The book is arranged in three parts: Part A presents the
most recent developments in the theory of Heyting algebras,
MV-algebras, quantales and GL-monoids. Part B gives a coherent and
current account of topos-like categories for fuzzy set theory based
on Heyting algebra valued sets, quantal sets of M-valued sets. Part
C addresses general aspects of non-classical logics including
epistemological problems as well as recursive properties of fuzzy
logic.
This book has a fundamental relationship to the International
Seminar on Fuzzy Set Theory held each September in Linz, Austria.
First, this volume is an extended account of the eleventh Seminar
of 1989. Second, and more importantly, it is the culmination of the
tradition of the preceding ten Seminars. The purpose of the Linz
Seminar, since its inception, was and is to foster the development
of the mathematical aspects of fuzzy sets. In the earlier years,
this was accomplished by bringing together for a week small grou ps
of mathematicians in various fields in an intimate, focused
environment which promoted much informal, critical discussion in
addition to formal presentations. Beginning with the tenth Seminar,
the intimate setting was retained, but each Seminar narrowed in
theme; and participation was broadened to include both younger
scholars within, and established mathematicians outside, the
mathematical mainstream of fuzzy sets theory. Most of the material
of this book was developed over the years in close association with
the Seminar or influenced by what transpired at Linz. For much of
the content, it played a crucial role in either stimulating this
material or in providing feedback and the necessary screening of
ideas. Thus we may fairly say that the book, and the eleventh
Seminar to which it is directly related, are in many respects a
culmination of the previous Seminars.
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.
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.
This book has a fundamental relationship to the International
Seminar on Fuzzy Set Theory held each September in Linz, Austria.
First, this volume is an extended account of the eleventh Seminar
of 1989. Second, and more importantly, it is the culmination of the
tradition of the preceding ten Seminars. The purpose of the Linz
Seminar, since its inception, was and is to foster the development
of the mathematical aspects of fuzzy sets. In the earlier years,
this was accomplished by bringing together for a week small grou ps
of mathematicians in various fields in an intimate, focused
environment which promoted much informal, critical discussion in
addition to formal presentations. Beginning with the tenth Seminar,
the intimate setting was retained, but each Seminar narrowed in
theme; and participation was broadened to include both younger
scholars within, and established mathematicians outside, the
mathematical mainstream of fuzzy sets theory. Most of the material
of this book was developed over the years in close association with
the Seminar or influenced by what transpired at Linz. For much of
the content, it played a crucial role in either stimulating this
material or in providing feedback and the necessary screening of
ideas. Thus we may fairly say that the book, and the eleventh
Seminar to which it is directly related, are in many respects a
culmination of the previous Seminars.
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.
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