Domain Approximable Concept Lattice Rough Sets And Topology

The contents of this paper consist of three main parts.The first part of this paper includes Chapterâ… ,â…¡,â…¢.In theoretical computer science, D.Scott initiated the domain theory which theoretical approximable to the semantics of programming languages at 1960s.At about the same time,in pure mathematics,J.D. Lawsom,K.H.Hofman discovered the theory of algebraic lattices,continuous lattices from the study of the structure of compact semi-lattice.From the different areas,many researchers' works have led to the growth of domain theory.They also obtained some equivalence representations for domain,such as information system,neighbourhood sys-tem,event structure,etc..1980,P.Martin-Lof propsed the intuitionistic type theory which aims to provide the foundation for constructive mathematics.Within P.Martin-LSf intuitionistic type theory, 1987,G.Sambin initiated the formal topology which includes algebraic Scott domain as a special case.1996,he also introduced the notion of information bases(unary formal topology),and proved the equivalence between the category of information bases with translations and the category of algebraic Scott domains with Scott continuous functions. So he obtaine a new,simple representation of algebraic Scott domains.Then he posed the problem;how much of domain theory can be generalized to formal topology,how to define a translation between two formal topologies which corresponds to the stable function? 2002,S.Negri investigated a continuous domain as formal topology.In Chapterâ… ,we briefly review the notions of domains,information systems,neigh-bourhood systems,which survey Preliminaries,provide a foundation for further study.In Chapterâ…¡,according to the definition of information bases given by G.Sambin, we investigate continuous information bases,continuous approximable relations,which correspond to continuous domains,Scott continuous functions.Our work reveals that continuous domains may be viewed as special cases of formal topology.This is a partial solution to G.Sambin's problem.In Chapterâ…¢,from the view of Chu space(formal concept analysis),Zhang,P. Htzler and Shen introducted the notion of approximable concepts,and proved the equiv-alence relation between algebraic lattices and approximable concept lattices.We also investigate the relation between algebraic information bases and approximable concept lattices.As a generalization of approximable concepts in the classical set theory,we construct the theories of approximable concepts,continuous lattices,algebraic lattices in fuzzy setting.The second part,i.e.,Chapterâ…£,Pawlak proposed the theory of rough sets to generalize the classical sets theory,and investigated the relation between the rough sets and topological space.Lin introduced the notion of neighborhood systems to handle such general situations.Many works have appeared recently.In Chapterâ…£,we introduce the notion of rough operators,and investigate the relation between rough sets and topology in fuzzy setting.The third part,i.e.,Chapterâ…¤,we investigate a kind of topology on complete lattices.In Chapterâ…¤,as a generalization of continuous lattices,1982,D.Novak introduced the notion of generalized continuous lattices.Following the work of D.Drake,W.J. Thron,J.B.Wright,S.Papert,for the construction of topological representation for a complete lattice,Prof.Deng investigated the generalized continuous lattices in 1996, and defined the notions of the maximal subset systems,lower(upper)homorphisms,and additive property,obtained the equivalence between the category of additivity generalized algebraic lattices and the category of T₀-topological spaces(2002).As a continuation, we construct Urysohn Lemma,Tietze Theorem,Stone compactification,which are new forms of the above results.