Fuzzy Semi-continuous Lattices And Their Categorical Properties

The classic domain theory was introduced in the early of 1970s, it prepared the mathematic ground for semantics of compute theory. In the classic domain theory, the simply order relations lack the quantitative information which is needed during the computing, that is, it can not reflect the amount of information which can be computed contained in the elements. The introduction of the fuzzy posets and the quantitative domain theory make up this shortfall. Thus,the study of the theory has aroused interests of many scholars. In this paper ,the fuzzy semiwaybelow relation is given in approach of fuzzy semiprime ideals, and based on it the concepts of fuzzy semicontinuous lattices and so on are introduced.Some properties of fuzzy semicontinuous lattices relating to the order, topology and category are studied. This thesis mainly includes four parts:Chapter One:Preliminaries knowledge. In this chapter, we give some prelim-inaries that related with classic domain theory, quantitative domain theory and category theory which we will be used throughout this paper.Chapter Two:Fuzzy senmicontinuous lattices. Firstly. the fuzzy semiwaybelow relation is defined .We show that the relation is different from the one which ap-plied in an article by giving an example .Based on it ,fuzzy semicontinuous lattices and fuzzy strongly continuous lattices are defined. Secondly, some basic properties of fuzzy semicontinuous lattices and fuzzy strongly continuous are disscused ,more over, some equivalent characterizations of fuzzy semicontinuous lattices are also ob-tained. Finally, the relation among fuzzy semicontinuous lattices. fuzzy continuous lattices and fuzzy strongly continuous lattices are discussed.Chapter Three:Fuzzy semiscott topology .Firstly, the definition of fuzzy semis-cott open sets and fuzzy semiscott open filters are introdueed .Some fundamendal properties of them are studied. Secondly. the relation between fuzzy semiscott topol-ogy and fuzzy scott topology is discussed. We obtained that the fuzzy semiscott topology is consistent with the fuzzy scott topology when the fuzzy poset is a fuzzy strongly continuous lattice .Finally. the fuzzy strongly semicontinuous function and fuzzy semiscott continuous function are defined and their relation is discussed. The conditions to characterize the fuzzy semiscott continuous function are given.We also obtained that the fuzzy semiscott continuous function is a fuzzy scott continuous function if the fuzzy poset is a fuzzy strongly continuous lattice.Chapter Four:The category properties of fuzzy semicontinuous lattices. Firstly. we obtained that the fuzzy semicontinuous lattices category has finite product. Sec-ondly, we discuss the conditions under which the function space of fuzzy strongly continuous lattices is fuzzy strongly continuous.