The Application Of Ω-categories In Quantitative Domain

Domain Theory produced from the double backgrounds of theoretical com-puter science and pure mathematics, is the foundation and core of theoretical computer science and play a decisive role in theoretical computer science. It prop-erly explains the semantics of programming by taking mathematics as the tool, using symbols and formulas. As its rich topology and order structure, and rela-tion closed with computational practice, it has been concerned by many scholars of computer science and mathematics, since the pioneering work of D.S.Scott and A.Ershov in the70’s.Quantitative Domain Theory is a new branch of Domain Theory and one of the hot topics in researches of current Domain Theory. Its main purpose is to provide quantitative model of concurrent language. Therefore, investigation on it will further promote the development of other disciplines of mathematics and theoretical computer science. At the same time, as the intersection and infiltration of it with other disciplines its research content is more abundant and its research significance is more greater.Ω-categories provide quantitative models for semantic of program languages, and thus becomes the main research field of Quantitative Domain Theory. In this paper, we explore the application of Ω-category theory in Quantitative Domain Theory. The details are as follows:(1) Establishing the algebraic Ω-categories:By discussing its basic properties such as algebras, projection, and product, we show that the category of continuous Ω-categories is finite closed. Then, we construct the algebraic Ω-categories and show that every continuous Ω-category is a continuous retraction of an algebraic Ω-category, and the continuous retraction of a continuous Ω-category is also a continuous Ω-category. Additionally, we obtain the result that the category of algebraic Ω-lattices is Cartesian closed.(2) Constructing Ω-completion of Ω-category:We discuss the universal prop-erty of a special completion-compact directed completion. Then, based on the join-and meet-completions, we construct the Ω-completion of Ω-category and prove that each Ω-completion of an Ω-category can be seen as the Dedekind-MacNeille completion of the union of a join-completion and a meet-completion of the original Ω-category. As a result of our study, it follows that all Ω-completions of an Ω-category can be characterized by Ω-formal contexts consisting of standard closure systems of the Ω-category.(3) Establishing the meet-continuity of Ω-categories:We show that the Q-category preserving sups of directed Ω-subsets and finite Ω-subsets is equivalent to that preserving sups of arbitrary Q-subsets. Then, inspired by the structure form of meet-continuity in Domain Theory, we discuss the quantitative problem for meet-continuity using two different methods. The fist, from the point of category, we es-tablish the meet-continuity and distributivity of Ω-semilattice and prove that every Ω-frame is equivalent to a meet-continuous and distribute Ω-lattice. Additionally, we investigated the products and Ω-closure systems on meet-continuous Ω-lattices. As a result of our study, it follows that the category of meet-continuous Ω-lattices with mappings preserving directed joins as morphisms is Cartesian closed. Sec-ond, from the point of fuzzy set, we also construct the meet-continuity of fuzzy semilattices by the operators preserving directed joins and similarly, discuss some properties of meet-continuous fuzzy semilattice and show that the category of meet-continuous fuzzy lattices with fuzzy Scott continuous mappings as morphisms is also Cartesian closed.(4) Establishing algebraic Ω-closure operators on Ω-categories:We propose the concepts of algebraic Ω-(?)-structures and algebraic Ω-closure operators. Then, we explore some basic properties of them and establish a correspondence from the algebraic Ω-(?)-structures to the algebraic Ω-closure operators. Finally, we introduce the Frink Ω-ideals and show that every Ω-category can be embedded into an algebraic complete Ω-category by the Yoneda embedding.This dissertation is supported by Hunan Provincial Innovation Foundation For Postgraduate.This dissertation is typeset by software (?)...