The Cardinal Functions On Continuous Domains And Some Cartesian Closed Subcategories Of SLP

The Cardinal Functions On Continuous Domains And Some Cartesian Closed Subcategories Of SLPLiu niAbstract Domain theory is an important study field of theoretical computer science. It had been established by some computer experts and mathematicians at nearly the same time by the end of 1960. Domain theory provides the mathematical foundation for denotation semantics of computer program languages. It is characterized by the close connection between order and topology, which makes it the common study field of both computer experts and mathematicians.Weight, character and density are generally called cardinal functions. They play an important role in general topology. Since domain is the close interaction of order and topology, we now consider these cardinal functions on continuous domains. Six distinguished computer experts including Scott defined the weight of continuous lattices in the first reference and proved that the three kind of weights 梬ei g ht of continuous lattices, weight of the related Scott topological space and weight of the related Lawson topological space, are consistent. Based on that, the definition of weight of continuous domains was given and some fundamental properties of it were discussed. In this paper, we first proved that those three weights about continuous domains are equal. After this, the definitions of local basis and character on continuous domains are given and some characteristic theorems of them are discussed. Finally the relations between three characters are obtained, that is, the character of continuous domains is equal to that of the corresponding Scott topological space and both not larger than that of the related Lawson topological space. We also have proved that these three cardinal functions are isomorphic invariable. Particularly, character is equivalent invariable. After this main topic, the characterization and definition of meet- continuous complete semi-lattices are given. By constructing two kinds of sup-semi lattices, the equivalence between the category of posets and the category of algebraic domains can be restricted to both the category of algebraic L-domains and that of Scott domains. To prepare for the next part, the Cartesian closed properties of three categories are given at the end of this part.Another main topic of this thesis is to discuss Cartesian closed property of some full subcategories of local complete semi-lattices with stable functions. The basic properties of stable functions are given firstly . Distributive dcpo is defined then and the stable function on this kind of dcpo is discussed in detail. The Cartesian closed properties of some distributive domains are obtained, such as the categories of distributive meet-continuous semi-lattices, distributive continuous complete semi-lattices, distributive algebraic complete semi-lattices (distributive Scott domains).Generally, we have that the categorical products are Cartesian products and the exponential objects are in fact the function spaces of stable functions under stable orders in any full subcategory of SLP. From this conclusion, we can judge whether a full subcategory of SLP is Cartesian closed. The last part of this thesis is to say something about the spectral theory of complete lattices and the sobrification of topological spaces. An important property of sobrification is given. By discussing the spectra of complete lattices, a theoretical foundation of sobrification of spaces is obtained. Thus we know that the category of sober spaces is a reflective subcategory of the category of all the topological spaces with continuous functions.