| With double backgrounds of theoretical computer science and pure mathematics, Domain Theory plays a fundamental role in the semantics of programming languages, and has close relations to topology, logic, algebra, category and some other mathematics disciplines. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way, and it is characterized by the close relation and interaction between orders and topologies. Quantitative Domain Theory (QDT for short) forms a new branch of Domain Theory and has undergone active research in the past three decades. QDT is concerned with models of computation that, in addition to qualitative information, allow also for the extraction of quantitative information—such as determining the speed of convergence or complexity of a program. At present several frameworks and approaches have been developed for QDT. Among them, the approach given by Ω-category theory attract a lot of attention.Ω-categories are a special kind of enriched categories, and they include ordered sets and generalized metric spaces as specific examples. This dissertation concerns several structures related to Ω-categories, and the application of Ω-category theory in QDT. The content of this dissertation includes three aspects. Firstly, we study algebras with consistent Ω-category structures. Secondly, for several problems exist in Ω-category theory, we study the intrinsic structure of Ω-categories. Thirdly, we explore the application of Ω-category theory in QDT. The structure of this dissertation is organized as follows:Chapter One:Preliminaries. This chapter gives some preliminaries that will be used throughout the thesis, including:basic concepts in domain theory, concepts and conclusions of Ω-category theory and the concept of fuzzy domain.Chapter Two:Ω-ordered algebraic structures. In this chapter, we study algebras with consistent Ω-category structures. Firstly, the concept of Ω-ordered semigroup and some examples are given. Secondly, homomorphisms and ideals in Ω-ordered semigroups are studied. Thirdly, based on Ω-adjunction, the notion of Ω-residuated ordered semigroup and several examples about it are given, and some properties about it are studied. Lastly, the relation between Ω-ordered algebraic structures and algebras with fuzzy equalities is established.Chapter Three:Basic structures about Ω-categories. Consisting of three sections, this chapter study several problems and structures about Ω-categories. In section one, we introduce the notion of many valued congruence relation in complete L-partially ordered sets. The relation between it and L-closure operator is established. We defined the quotient of congruence on complete L-partially ordered sets, and proved that the image of a surjective homomorphism of complete L-partially ordered sets is isomorphic to the quotient the congruence relation induced by that homomorphism. In Section two, we study contravariant Galois connections on Ω-categories. The representations of complete Ω-categories and tensor product of complete Ω-categories by contravariant Galois connections are given. And, it is proved that contravariant Galois connections on Ω-categories can be represented by certain Ω-valued relations. Section three is concerned with many valued topologies based on Ω-categories. A stone like dual is established between the category of strong L-topologies and the category of L-frames.Chapter Four:Products of categories of fuzzy domains. In this chapter, first of all we study the relationships among three kinds of completeness of L-partially ordered sets. It is proved that an L-partially ordered set is complete if and only if it is both finite join complete and directed complete. Secondly, we study the relation between fuzzy domain and crisp domain. The methods and conditions of constructing them from each other are given. Thirdly, we give the concrete formation of fuzzy way below relation in product of fuzzy domains. Furthermore, we prove that the category of pointed cotensored fuzzy domains and the category of fuzzy continuous lattices have product.Chapter Five:Cartesian closedness of the category of (algebraic) fuzzy continuous lattices. This chapter is devoted to search for Cartesian closed subcategories of fuzzy do-mains. It is mainly proved that the category of fuzzy continuous lattices and the category of algebraic fuzzy continuous lattices are Cartesian closed. In section one of this chapter, we recall some properties of the images of fuzzy domains under several fuzzy Scott con-tinuous projection operators. It is mainly proved that the image of a fuzzy domain under a fuzzy Scott continuous projection operator remains to be a fuzzy domain. Furthermore, we study the continuity of function spaces of fuzzy domains, based on the results about products of the category of fuzzy continuous lattices given in Chapter four, we prove that the function spaces of fuzzy continuous lattices are fuzzy continuous lattices. Therefore, we prove that the category of fuzzy continuous lattices is Cartesian closed. After intro-ducing the definition and some properties of algebraic fuzzy domains, we prove that the category of algebraic fuzzy continuous lattices have products and function spaces and the category of algebraic fuzzy continuous lattices is Cartesian closed. |
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