Theoretical Research Into Bases Of Fuzzy Domains And Its Generalization

The concept of continuous lattices was found by Scott in1970’s, where it was applied in a topological manner as a mathematical tool in computer sciences (domain theory). Although it has appeared in other fields of mathematics as well, such as general topology, category theory, logic, this notion was defined in purely order theoretical terms and now has become one, which has been applied in almost all references. On the other hand, ever since fuzzy order was proposed by Zadeh in1971, various kinds of fuzzy orders have been introduced and have been widely applied in computer sciences. In order to investigate L-Fuzzy domain via fuzzy sets, based on frames, Zhang and Fan introduced fuzzy partial order and L-Fuzzy directed subset, then presented the concept of L-Fuzzy domain. But the definition of this kind of fuzzy directed subsets looks relatively complex. For this reason, Lai and Zhang, Yao introduced another kind of fuzzy directed subsets and studied their continuity respectively, which looks relatively simple and accessible. In this paper we will still focus on the research of the later. We were mainly from the following aspects to elaborate.The study of fuzzy local bases. The definition of fuzzy local bases was present-ed to investigate fuzzy domain. After about20years of development, quantitative domain theory forms a relatively perfect theory system and has undergone active research. But their research mainly focused on the properties of fuzzy domains. To this end, we introduced the definition of fuzzy local bases to take advantage of local properties to characterize the overall nature. It was proved that fuzzy dcpos are fuzzy domains if and only if for each point, there exists a fuzzy local basis of it. Due to the important role in quantitative domain theory, the interpolation proper-ty was widely studied, where only necessary conditions were given. We confirmed that the sufficient conditions also hold in this paper.The investigation of fuzzy bases. Domain can provide models for various type-s of programming languages that include imperative, functional, nondeterministic and probabilistic languages. When domains appear in theoretical computer sci-ences, one wants them to be objects suitable for computation. In particular, one is motivated to find a suitable notion of a recursive or recursively enumerable do-main. To this end, we proposed the notion of a fuzzy basis to characterize fuzzy domains. It is different from the crisp case, if it has a fuzzy basis, then there exists a fuzzy lower set of it. Furthermore, the concept of a fuzzy algebraic domain was introduced, and a relationship between fuzzy algebraic domains and fuzzy domain-s was discussed from the viewpoint of fuzzy bases. We also gave an application of fuzzy bases, where the image of a fuzzy domain can be preserved under some special kinds of fuzzy Galois connections.Theoretical research into fuzzy Z-continuous posets. To introduce higher type variables into recursion equations, we presented the notion of fuzzy Z-continuo posets as a generalization of fuzzy domains. An important research direction of domain is to generalize it. For this purpose, Wright, Wagner and Thatcher in IBM Lab introduced the notion of subset systems, replacing the system of all directed subsets by other types of subsets. We gave the definition of fuzzy sub-set systems, then presented the notions of fuzzy Z-continuous posets and fuzzy strongly Z-continuous posets. We introduced the concept of fuzzy Z-complete closure systems and associated a fuzzy Z-continuous closure operator with a fuzzy Z-complete closure system. We proved that each fuzzy Z-complete closure sys-tem of a fuzzy Z-continuous poset is fuzzy Z-continuous. The notion of fuzzy Z-clgebraic posets was given, then some algebraic properties of such a structure were studied. An extension theorem was presented for extending a fuzzy monotone map defined on the Z-compact elements to a fuzzy Z-continuous map defined on the whole set.A representation of fuzzy completely distributive lattices. When Belohlavek discussed the relationship between the concept analysis and complete lattices, he originally introduced the fuzzy version of complete lattices as a generalization of ordinary complete lattices. Zhang and Xie were devoted to a systematical study on fuzzy complete lattices and Dedekind-MacNeille completions. Lai and Zhang presented the definition of completely distributive Ω-lattices. Yao pointed out that a fuzzy completely distributive lattice can be obtained via the approximation of a fuzzy well-below relation. We focused on the later in this paper, we introduced the concept of a fuzzy cut set. We proved that the family of all fuzzy cut sets can form a fuzzy completely distributive lattice. We gave the definition of approximate elements, and also presented some equivalent characterizations for fuzzy completely distributive lattices.This dissertation is typeset by software LATEX2ε...