The Representations Of Some Subclasses Of Domains Via Information Systems And Closure Systems

In the 1970’s, Turing Award Winner Dana Scott proposed Domain theory, it is significant in denotational semantics of programming languages. In the field of pure mathematics, Lawson, Stralka, etc. defined a kind of complete lattices with special properties in order structure researching. They soon found that this complete lattice were exactly the continuous lattices introduced by Scott. This result attract researchers wide attention, stimulate the development of Domain theory rapidly. Domain theory is an interdisciplinary field of computer science and topology, order theory and category theory, it is an abstract research field. After more than 40 years of development, this theory obtained rich achievement, but also some problems have been left.One important topic of Domain theory is concrete representation of every sub-class of domain by various modalities, such as, the description by means of topologies, closure systems, information systems etc. In 1982, Scott introduced the notions of infor-mation system and approximate mapping. Larsen and Winskel proved that the category of information systems is equivalent to the category of Scott domains and continuous functions, this means the information system representation of Scott domain has been found. Later on, Hoofman gave the information systems that represent bounded com-plete continuous domains. As the problem of information system representation for continuous domain, Professor Spreen and Luoshan Xu etc. solved this problem until 2008, they called this notion as general continuous information system. On this basis, they obtained the general algebraic information system as representation of algebraic do-main.The category formed by domains and continuous function is not Cartesian closed, so finding maximal cartesian closed full subcategories of the continuous domains is the most important research work in Domain theory. L-domains were independently intro-duced by Coquand and Jung. Jung showed that L-domains form a maximal cartesian closed full subcategory of the continuous domains. Therefore, L-domains have research significance. On the base of general continuous information systems, Spreen gave an information system representation of L-domains. However, this information system has many rules, it is complicated.This paper proposed two new information systems, and proved that they are infor-mation system representations of algebraic domains, algebraic L-domains respectively. The first concept is algebraic information system, it is an representation of algebraic do-main. This information system is on the base of compact elements which are a particular basis of algebraic domains, so rules are concise. On the notion of algebraic information system, we introduced the appropriate mappings as morphisms of them, then we proved that all of algebraic information system and appropriate mappings informed a category. Furthermore, by construction functors between algebraic domain category and algebra-ic information system category, the categorical equivalence are obtained. On the basis of algebraic information system, a new notion of algebraic L-information system are introduced. Similarly, appropriate mappings are appropriate morphisms of algebraic L-information system, the equivalence of algebraic L-information system category and algebraic L-domains are proved in this paper.The next important work of this paper is the information system representation of L-domains. Because L-domain has local least upper bound, this notion is a difficulty of information system representation, we found that if we use arbitrary base to con-struct information system, the complexity of the representation is unavoidable. So we considered a particular basis, presented the notion of L-information system. It provided a concrete representation of L-domain with less rules, so it is concise. Then, we also given appropriate appropriate mappings as morphisms of L-information system, and we prove the equivalence of L-information system category and L-domain category.The description of order structure by sets is an concerned research topic. In the early 20th century, Stone found that any Boolean lattice is isomorphic to the family of the clopen sets of a totally disconnected topological space which is also called Stone space. Priestley generalized this result to bounded distribute lattice, obtained Priestley space corresponding to bounded distribute lattice. In domain theory, a classic conclusion is that algebraic closure systems can be used to reconstruct algebraic lattices。Another well-known result is that every continuous lattice is isomorphic to the image of a pro-jective operator which preserves directed sups on a powerset. Recently, Lankun Guo and professor Qingguo Li presented the notion of F-augmented closure spaces, which provides an alternative way to represent algebraic domains. Based on the algebraic clo-sure system, the notion of singly separated closed subsets is introduced, which provides a new approach to reconstruct algebraic domains. Then, a special type of algebraic closure spaces named algebraic L-closure spaces is proposed, which give us a tool to represent algebraic L-domains. Furthermore, the notion of algebraic approximate map-pings is introduced, which serves as the appropriate morphism for algebraic closure spaces and algebraic L-closure spaces. On the viewpoint of category, we finally ob-tain the equivalence between algebraic closure spaces category (respectively, algebraic L-closure spaces) with algebraic approximate mappings as morphisms and algebraic do-mains category (respectively, algebraic L-domains) with Scott continuous functions as morphism.