Domain Representation Theory Of Several Generalized Fuzzy Metric Spaces

The theory of fuzzy metric spaces is an important branch of fuzzy set theory.One type of fuzzy metric spaces,which is based on continuous t-norms,received the highest attention due to its background in probabilistic theory and various application in gen-eralized measure theory,the theory of topological spaces,image processing and etc..Recently,as the theory of fuzzy metric spaces developed rapidly,some notions of gen-eralized fuzzy metric spaces were introduced.Domain theory and the theory of metric spaces are two central tools for the study on denotational semantics of programming language.There are plenty of research,which have shown that there is a deep relation-ship between Domain theory and the theory of metric spaces.Naturally,research on the connection between Domain theory and the theory of generalized fuzzy metric spaces attracted much attention,and became an important topic.However,the research meth-ods for classical results do not act upon the fuzzy setting.New tools are required in the study of the relationship between Domain theory and the theory of generalized fuzzy metric spaces.The aim of this paper is to study the application of Domain theory to several types of generalized fuzzy metric spaces,which include fuzzy ultrametric spaces,fuzzy quasi-metric spaces and fuzzy partial quasi-pseudo-metric spaces.The tools associated with Domain theory that are used in this paper contain standard closed balls and formal balls.(1)In a fuzzy ultrametric space(X,M,?),by using the tool of standard closed balls CS[X],we prove that the following conditions are equivalent:(1)(X,M,?)is standard complete;(2)The poset(CS[X],(?)M)constitutes a Scott domain;(3)The poset(CS[X],(?)M)constitutes a dcpo.Besides,the results established in this paper also show that the characterization of standard completeness of fuzzy ultrametric spaces can also be obtained through formal ball method.Finally,we obtain that each fuzzy ultrametric space(X,M,?)has a computational model of Scott domain.(2)For a fuzzy quasi-metric space(X,M,?),we introduce the concepts of Yoneda T-completeness and Yoneda S-completeness.The main results are:(1)If a fuzzy quasi-metric space(X,M,?)is Yoneda T-complete or Yoneda S-complete,then the related posets of formal balls constitute dcpos;(2)Conversely,if the associative poset of formal balls,which is equipped with the partial order relation(?)MT((?)MS),constitutes a dcpo,then(X,M,?)is T-complete(S-complete).For a fuzzy metric space(X,M,?),we prove that the following conditions are equivalent:(1)(X,M,?)is Yoneda T(S)-complete;(2)(X,M,?)is T(S)-complete;(3)The associated poset of formal balls is a dcpo.(3)Besides,this paper also focuses on the GV-fuzzy quasi-metric spaces.We sys-tematically investigate the connections between several types of completeness on GV-fuzzy quasi-metric spaces.We investigate relationship between standard Yoneda com-pleteness,standard completeness and the poset of formal balls consisting a dcpo.Based on different technics with Chapter 4,we obtain the similar results as mentioned in fuzzy quasi-metric spaces.(4)Finally,we introduce the concept of fuzzy partial quasi-pseudo-metric spaces and consider its representation problem by characteristic family.The main results of this paper have solved an open problem in a relevant paper.