Fuzzy Domain And Its Characterizations

By intruducing a fuzzy partial ordering of lattice value (in this dissertation, we denote by L the completely distributive lattice), we build a new structure of theory of L-fuzzy Domain and use ideas of the four nested sets and the corresponding level cut sets in [1] - [4] and obtain a lot of equivalent characterizations about L-fuzzy Domain and related conceptions. Furthermore, we make more deep discussion than [5] about the corresponding Alexandroff topology. Our main results are:In Chapter one, we intruduce the preliminary knowledge of L-fuzzy Topology and Domain.In Chapter two, we define L-fuzzy Poset and give the decomposition and equivalent characterization of L-fuzzy Poset, L-fuzzy monotone mapping by means of four level cut sets.In Chapter three, we intruduce concepts of L-fuzzy directed subset, directed union, L-fuzzy Domain, L-fuzzy lower set, L-fuzzy ideas, L-fuzzy base, give its equivalent discription, prove the equivalent relation between L-fuzzy concepts and classical conceps, and discuss properties of (?)_L approximation and continuous L-fuzzy Domain.It is proved in [5] that under the condition of one cut set the generalized Alexandroff topology on L-fuzzy preorder set is a generalization of Alexandroff topology on usual preorder set, and that a generalized Alexandroff topology on L-fuzzy preorder set can be obtained by the union of a nest of Alexandroff topology. In Chapter four, we show the same conclusions hold under the condition of the other three cut sets.