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.
