A Research On Categorical Dualities Between Certain Kinds Of Fuzzy Posets

With a strong background of theoretical computer science and pure mathematics, Domain Theory plays a fundamental role in the semantics of programming languages, and has close relations to topology, logic, algebra, category and some other mathematics disciplines. Quantative Domain Theory, experienced a rapid development in the past four decades, forms a new branch of Domain theory. In this paper, we mainly use approaches of Ω-categories and fuzzy partial orders to generalize the catergory dualities into fuzzy situation. For details,Content One:Fuzzy G-ideals. Based on the fuzzy partial order theory, we firstly propose a definition of fuzzy G-ideals, and prove that the set of all fuzzy G-ideals of two fuzzy complete lattices is also a fuzzy complete lattice with respective to the subsethood degree. It mainly proves that there is a one-to-one relationship between fuzzy G-ideals and fuzzy Galois connections.Content Two:Fuzzy Scott topology. First, it is shown that a fuzzy Scott topology is not necessary a fuzzy topology. Second, the relationship between Scott topologies and Scott continuous maps is investigated; Furthermore, L-possibility computations are introduced and then an equivalence between their denotational semantics and their logical semantics are established.Content Three:Duality of fuzzy Domains. In this part, we first investigate and study the relations among various types of functions in the context of fuzzy Galois connections. Then the categorical dualities between certain kinds of fuzzy posets are established.