Some Generalizations And Applications Of Domain Theory

In the study of domain theory, rough set theory and modal logic, structures of order, topology and algebraic infiltrate and affect mutually. As well known, we can use the results and methods of domain theory to study rough set and modal logic, because in domain theory, the concepts and ideals of order, approximation and logic can be transformed into each other and unified. In this paper, we do the work in the following two aspects.First, an important content of domain theory is to promote the theory of continuous lattices as much as possible to more general order structures. The second to the fourth chapter of this article is some of the latest work in this area.In the second chapter, concepts of quasi-finitcly separating maps and quasi-approximate identities arc introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are:(1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains are all QFS-domains;(2) QFS-domains are compact in the Lawson topology:(3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is com-pact in the Lawson topology;(4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains.In the third chapter, the concepts of semiprime sets and semicontinuous dcpos are introduced. Basic properties of semicontinuous dcpos are discussed. Semicontinuous maps and some intrinsic topologies-the semi-Scott topology and the semi-Lawson topology on semicontinuous dcpos are investigated.In the fourth chapter, the concept of natural way-bclow relations is intro-duced for natural poscts, and the definition of naturally continuous natural poscts is given. It is proved that on a naturally continuous natural poset, natural way-below relation has the interpolation property and the lattice of natural Scott open sets is a completely distributive lattice. In terms of S*-convergence, natural way-below relations, natural Scott topologies and natural continuity of natural posets are characterized.The second aspect of the work we do is to try to use the methods of order and topology to discuss issues related to rough sets and modal logic. The contents of this fields are discussed in Chapters V and VI of this article.Chapter V is devoted to investigate special elements and order structures of rough sets from the view of domain theory. Completely compact elements and atoms of R are represented. In terms of the representations, it is proved that the class R is isomorphic to a complete ring of sets, consequently R is a completely distributive algebraic lattice. An example is given to show that R is not atomic and a sufficient and necessary condition for R being atomic is thus given. Another purpose of this chapter is to examine the lattice structures of probabilistic rough sets. The concept of a rough membership function in a probabilistic approximation space are introduced. It is proved that the family of all rough membership functions in a probabilistic approximation space forms a stone lattice. In terms of rough membership function, the notions of α-lower approximation operator Rα and β-upper approximation operator Rβ are de-fined and studied. Then the notion of probabilistic rough sets in probabilistic approximation spaces is proposed. It is proved that the family of all probabilistic rough sets forms a complete Stone lattice, generalizing the corresponding result for traditional rough sets in Pawlak’s sense.In the last chapter, for close linkages between order theory and topology, semantic models for modal logic system S4and S5are studied. It is well known that in semantics of modal logic, one can define different concepts of models in terms of different mathematical structures. Typical models are defined by rela-tional structures, and a kind of topological models is given by using topological structures. Practically, different kinds of models have their advantages and disad-vantages. Obviously, finite relational models and some special finite topological models are most convenient ones. So in this chapter, some more linkages of rela-tional models and some special topological models are obtained, and meanwhile equivalences of some classes of (finite) models are given. As a corollary, Question9.1.66in "Non-classical Mathematical Logic and Approximate Reasoning (2nd edition)" is affirmatively answered.