L-fuzzy Domain Theory,

In this thesis,based on L-fuzzy quasi ordered set introduced in [16],generalized Alexandroff topology on an L-fuzzy quasi ordered set is defined,a domain theory with respect to this L-fuzzy quasi order,which is called L-fuzzy domain theory,is built. The primary studies in this paper are the following:(1) We define a generalized Alexandroff topology on an L-fuzzy quasi ordered set which is a generalization of the Alexandroff topology on an ordinary quasi ordered set,prove that the generalized Alexandroff topology on an L-quasi ordered set (X,e) can be obtained by the join of a family of the Alexandroff topologies on it,a topology on any topological space can be represented as a generalized Alexandroff topology on some L-quasi ordered set,and the generalized Alexandroff topologies on L-fuzzy quasi ordered sets are generalizations of the generalized Alexandroff topologies on generalized ultrametric spaces which are defined by J.J.M.M.Rutten etc.(2) By introducing the concepts of the join of L-fuzzy set on an L-fuzzy partial ordered set with respect to the L-fuzzy partial order and L-fuzzy directed set on an L-fuzzy quasi ordered set (with respect to the L-fuzzy quasi order),we define L-fuzzy directed-complete L-fuzzy partial ordered set (or briefly,L-fuzzy dcpo or L-fuzzy domain) and L-fuzzy Scott continuous mapping,prove that they are respectively generalizations of ordinary dcpo and Scott continuous mapping,when L is a completely distributive lattice with order-reversing involution,the category L-FDom of L-fuzzy domains and L-fuzzy Scott continuous mappings is isomorphic to a special kind of the category of V-domains and Scott continuous mappings,that is,the category L-DCQum of directed-complete L-quasi ultrametric spaces and Scott continuous mappings,and when L is a completely distributive lattice in which 1 is a molecule,L-fuzzy domains and L-fuzzy Scott continuous mappings are consistent to directed lim inf complete categories and lim inf continuous mappings in [59].(3) We build the L-fuzzy ideal completion of L-fuzzy quasi ordered set,that is,we prove that the set of all L-fuzzy ideals on an L-fuzzy quasi ordered set with proper extension mapping is an L-fuzzy domain,and an L-fuzzy monotone mapping from an L-fuzzy quasi ordered set to an L-fuzzy domain can be extend to an L-fuzzy Scott continuous mapping. We define generalized Scott topology on an L-fuzzy domain,prove that it is a generalization of Scott topology on ordinary domain,and an L-fuzzy monotone mapping is an L-fuzzy Scott continuous mapping if and only if it is continuous with respect to the generalized Scott topologies,which means that topological continuity is identical to limit continuity.(4) We construct the concepts of stratified approximations and base in an L-fuzzy domain,and define uniformly continuous L-fuzzy domain and algebraic L-fuzzy domain by means of the concept of base for L-fuzzy domain,prove that they are generalizations of ordinary continuous domain and algebraic domain,when L is a completely distributive lattice in which 1 is a molecule and the minimal mapping on L preserves finite meet,the stratified approximation relations on a continuous L-fuzzy domain satisfy the property of stratified interpolations,and is consist of a base for the generalized Scott topology on a continuous L-fuzzy domain (X,e),and every continuous L-fuzzy domain can be seen as an L-fuzzy Scott continuous retraction of some algebraic L-fuzzy domain.(5) We build representation theories for algebraic L-fuzzy domains,that is,we prove that when L is a completely distributive lattice in which 1 is a molecule and the minimal mapping on L preserves finite meet,the category L-AlgFDom of algebraic L-fuzzy domains and L-fuzzy Scott continuous mappings which preserve L-approximation-orders is equivalent to the category L-CCPPos of Cauchy complete L-fuzzy partial ordered sets and L-fuzzy monotone mappings,and the category L- AlgFDom of algebraic L-fuzzy domains and L-fuzzy Scott continuous mappings is equivalent to the category L-CCFPosafl of Cau...