The Research On The Representations Theory Of Continuous Lattice And FS-domain

In 1969,D.S.Scott proposed Domain theory,the purpose is to provide the mathe-matical foundation for denotational semantics of programming languages.Mutual trans-formation and infiltration of order,topology and logic are the basic features of this the-ory.Therefore,Domain theory has become a field of common interest by topology and computer science researchers.Continuity is a very important concept in domain theory,the characterization and representation of special types of domains are always important research topics.Many approaches,such as information systems,families of subsets of a set,topological spaces,formal concepts,etc as the representation of domains draw many scholars attention.As is well-known,the categories of continuous domains and algebraic domains are not Cartesian closed.In 1989,A.Jung proved that there are four maximal Carte-sian closed full subcategories in categories of continuous domain and algebraic domain,category of continuous L-domains,category of algebraic L-domains,category of FS-domains and category of bifinite domains(algebraic FS-domains).In 2014,Wu,Li and Zhou introduced the notion of the algebraic L-information system which captures al-gebraic L-domain.Guo,Li and Yao introduced a special type of formal context,and provided a concrete representation of algebraic L-domains via formal concept analysis as an efficient approach.Spreen,Luo and Mao proposed a new information systems as a representation of FS-domains,but the set Con was restricted to a countable collection of finite subsets.Based on these works,the main context of our theses can be divided into two parts:One is to remove the countability restriction of the Con in Spreen's paper,by taking our abstract base as a bridge to establish the representations of FS-domains and bifinite domains,the other is to investigate the representations of continuous lattices,FS-domains and bifinite domains by means of closure spaces,by incorporating an addi-tional structure into a given closure space in an appropriate way.The detailed content is listed below:In Chapter 1,a general survey of domain theory and innovation points is given in our paper.In Chapter 2,the basic concepts and results of continuous domains,domain cate-gories are supplied.In Chapter 3,the concepts of FS-bases,FS-information systems have been provid-ed and the representations of FS-domains by information systems have been presented.Meanwhile,the concepts of BF-bases,BF-information systems have also been intro-duced,and we obtain the result that the category of bifinite domains is equivalent to that of BF-information systems.In Chapter 4,we establish the link between continuous lattices and closure s-paces.By generalizing the notion of algebraic closure space to continuous closure s-pace,we show that continuous lattices can be represented by continuous closure spaces.We also introduce the notion of approximable mappings between continuous closure spaces,which produces a category equivalent to that of continuous lattices with Scott-continuous functions.In Chapter 5,by adding a special structure into a given closure spaces,we introduce the concept of FS-closure spaces and obtain a necessary and suffucient condition for the continuous domain to be a FS-domain.Furthermore,we prove that the category of FS-closure spaces with approximable mappings as morphisms is equivalent to that of FS-domains with Scott-continuous functions as morphisms.In Chapter 6,we investigate the representation of bifinite domains by means of closure spaces.We prove that every bifinite domain can be obtained as the set of F-closed sets of some BF-closure space under set inclusion.Furthermore,we obtain that the category of bifinite domains and Scott-continuous functions is equivalent to that of BF-closure spaces and F-morphisms.