Fuzzy Z-domain And Its Associated Categorical Properties

Domain theory has both theoretical computer science and mathematica research background dual. In a classical poset, only qualitative information between elements can be given, without quantative information for computing. So a classical poset can't express an element has how much information for computing. Fuzzy poset and quantitative Domain can remedy this shortcoming. Subset system opened up the research space of Domain theory. In this paper, the concept of subset system will be extended to fuzzy theory, and a new concept of fuzzy Z-subset system is introduced and based on it the concepts of fuzzy Z-Domain and so on are defined. Some properties of fuzzy Z-Domain relating to order, topology and category are studied.The arrangement of this thesis is as follows:Chapter One:Preliminaries. In this chapter, the basic concepts and existing results of the theories of Domain, topology, category and fuzzy set which will be used throughout the thesis are given.Chapter Two:Fuzzy Z-Domains. Firstly, the concept of fuzzy Z-subset sys-tem is defined. Some basic properties about it are studied. Secondly, the LZ-way below relation is defined and based on it the concepts of LZ-Domain and so on are introduced. Some basic properties about LZ-Domain are given. Finally, a stratified L-fuzzy convergence structure Rz is defined. Thus it induces an L-topology, called the fuzzy Scott topology of LZ-Domain, named LZ-Scott topology for short.Chapter Three:Fuzzy Consistently Domain. Firstly, the LC-way below relation is defined, and the concept of LC-Domain is established. An equivalent condition of a fuzzy consistently Domain is given. Secondly, the fuzzy topology of LC-Domain is discussed, and we proved an equivalent condition of a fuzzy consistently Scott open set. Thirdly, fuzzy consistently Scott continuous maps between LC-Domains are studied. It is showed that a map is fuzzy consistently Scott continuous if and only if the premage of every fuzzy consistently Scott closed set is also a fuzzy consistently Scott closed set. Finally, the SLC-convergence of a stratified L-fuzzy filter of LC-Domain is discussed. It is proved that a fuzzy consistently directed complete poset is an LC-Domain if and only if SLC-convergence equivalent to by fuzzy consistently Scott topology convergence for every stratified L-fuzzy filter.Chapter Four:The category properties of fuzzy consistently directed complete poset. It is proved that, the set of all fuzzy consistently ideals of fuzzy consistently directed complete poset given the appropriate fuzzy partial order can form a fuzzy consistently directed complete poset. In the case of L is a complete Heyting algebra, it is showed that the category LCDCPO is cartesian closed.