| Abstract With the rapid development of computer science, people are paying more and more attention to the research about its mathematical foundations which have been the area of common intersets of mathematicians and computer scientists. Domain theory and quantale theory established in the early1970's and1980's, re-spectively, are just two very important cross fields. They developed independently, but they are based on order theory from the viewpoint of the common mathemati-cal foundation. Meanwhile, they make the close relationship with topology, algebra, category, logic and so on. Although domain theory and quantale theory have differ-ent research objects and characteristics, they are mutual penetration and influence in some aspects, such as the applications of quantale theory to quantitative domain theory. Since2000, fuzzy set theory have been applied to the quantitative domain theory, forming fuzzy domain theory. The first part of this thesis is to further inves-tigate fuzzy domain theory. The second part is to study the fuzzification of quantale theory. The structure of this thesis is organized as follows:Chapter One: Preliminaries. In this chapter, some concepts and results will be used in this paper are given.Chapter Two:Join-completions of fuzzy posets. Firstly, the definition of join-completions of fuzzy posets is given. It is proved that the join-completions of a fuzzy poset (X. e) up to equivalence are completely determined by consistent fuzzy closure operators on Lx. Secondly, the universal property of join-completions is studied. At last, a categorical characterization of the Dedekind-MacNeille completion for a fuzzy poset is presented.Chapter Three:Φ-continuous fuzzy posets. Firstly, some related properties of weight classes are discussed. Some equivalent characterizations of preserving fuzzy joins mappings between fuzzy complete lattices are given. Secondly, some related properties of Φ-continuous fuzzy posets are obtained. We discuss some particular mappings under which the images of Φ-continuous fuzzy posets are still Φ-continuous fuzzy posets. At last, fuzzy SΦ-convergence of L-filters is studied.Chapter Four:Φ-algebraic fuzzy posets. Firstly, the concept of Φ-algebraic fuzzy posets is introduced. Some properties of Φ-algebraic fuzzy posets are ob-tained. Secondly, we discuss some properties of Φ-homomorphisms and give a classification theorem of fuzzy posets, which generalizes Hoffmann's classification theorem to the framework of saturated class of weights. At last, bases and weights on Φ-complete fuzzy posets are studied. The relations between the category of Φ-algebraic fuzzy posets and the category of fuzzy posets are discussed. It is proved that ΦAFPOSH is equivalent to FPOS. It is also proved that FPOID is dual equivalent to ΦAFPOSM·Chapter Five:Fuzzy quantales. Firstly, we introduce the concept of fuzzy quantales by means of fuzzy Galois connections and give some examples of fuzzy quantales. The quantic nucleus and quantic conucleus on fuzzy quantales are stud-ied. Secondly, the concept of Girard quantales is introduced. It is proved that L-quantic nucleus and L-ideal conucleus are one-to-one corresponding. Finally, fuzzy quantale completions of fuzzy ordered semigroups are discussed. It is shown that up to isomorphism, the fuzzy quantale completions of a fuzzy ordered semigroup (S1,·, e) are completely determined by topological fuzzy closure operators on LS.Chapter Six:The category of fuzzy quantales. In this chapter, we prove that the category of fuzzy frames introduced by Yao is a full reflective subcategory of the category of fuzzy quantales. Secondly, it is shown that the category of fuzzy quantales is isomorphic to the category of L-algebras. Finally, we give the structures of the limit and the inverse limit of the category of fuzzy quantales.
|
PDF Full Text Request