Several Classes Of Sub - Categories Of Fuzzy DCPO Categories

Domain theory is founded by Scott in the late1960s. Its purpose is to provide denotational semantics for the functional languages of computer. After years of development, Domain theory has achieved a very rich research achievements. To support all kinds of operations of calculation function, the corresponding domain category is often required to be cartesian closed, thus the research of cartesian closedness occupies very important position in Domain theory. Since2000, Fan Lei and Zhang Qiye have proposed the concept of L-fuzzy posets and expanded many results in the classical domain to fuzzy domain, thus formed the basic theories of fuzzy domain. Main results about the cartesian closedness of fuzzy domains are as follows. Yao Wei proves that FDCPO is cartesian closed and Liu Min proves that the category of fuzzy continuous lattices and the category of algebraic fuzzy continuous lattices are cartesian closed. But it is still not known that whether the category of fuzzy domains or the category of algebraic fuzzy domains which are two important full subcategories of FDCPO is cartesian closed.On one hand, this thesis is to give two cartesian closed full subcategories of FDCPO, which are the category of bounded complete fuzzy dcpos and the category of tensor complete fuzzy dcpos. On the other hand, the properties of finite products, equalizers of the category of fuzzy domains and the category of algebraic fuzzy domains are discussed. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we recapitulate the basic concepts and results of L-ordered sets and category theory which will be used throughout this thesis.Chapter Two:Cartesian closed subcategories of the category of fuzzy dcpos. In this chapter, firstly, we give corresponding conclusions about the cartesian closedness of FDCPO and a necessary and sufficient condition for the cartesian closedness of the full subcategories of FDCPO is given. Secondly, the concept of bounded complete fuzzy dcpos and tensor complete fuzzy dcpos are introduced and we prove that these two categories are cartesian closed, that is these two cartesian closed full subcategories of FDCPO. Finally, we give the definition of fuzzy complete semilattices and strong fuzzy complete semilattices and their relations with bounded complete fuzzy dcpo are discussed. Chapter Three:Some properties of fuzzy domain. In this chapter, the proper-ties of fuzzy domains and algebraic fuzzy domains are discussed. Firstly, we give an equivalent characterization of fuzzy domain, then we prove that the category of algebraic fuzzy domains has finite products, so we can get that the category of bounded complete fuzzy domains and the category of bounded complete algebraic fuzzy domains have finite products and we prove those two categories have equal-izes. Finally, the concept of fuzzy semilattice is given and the properties of fuzzy semilattice under some special maps are discussed.