Researches On Some Problems In Domain Theory

Domain is the theory with relation to topology theory,lattice theory and category theory,but also has a computer background.Since the last century in 70s,it attract-ed many scholars in mathematics and computer program linguists.After decades of development,the party in the face of the Domain structure itself has a more in-depth understanding,from continuous lattices to continuous dcpos,to continuous posets,then to quasi continuous posets,meet continuous posets,their properties and the relations with topology and categories are more mined.On the other hand,Domain theory and information system,formal concept,rough set,fuzzy set theory are more closely linked.Of course,there are still many problems to solve it.In this thesis,we construct the operator theory on general posets.In classical rough theory,the most important one is the upper and lower approximation operators.Using them,we can get qualitative analysis of uncertain information.The classical rough ap-proximation operators are established through the equivalence relation,but general order relation on posets and approximation relation are not equivalence relation.Therefore,we use the auxiliary relation on posets to establish another form of approximation oper-ators.The operators with the continuity of poset,the interpolation property of auxiliary relation,and the Scott topology on posets are closely linked.We obtain the main re-sult that:the continuity of poset can be characterized by the approximation operators.Finally,by using the operators,the Scott closure operator on the continuous poset are discussed,and the element composition of Scott closure operator is obtained.The paper also carried on the three open problems in Domain theory.1,the repre-sentations of continuous lattice,and completely distributive lattice by means of family of subsets.2,the necessary and sufficient conditions of the 02-convergence to be topo-logical on posets.3,FS-domain and RB-domain are equivalent or not.Sets are one of the most important tool in mathematics,the representations of spe-cial lattices by the family of subsets have always been an important research topic in Domain theory.We obtain a sufficient condition for a family of subsets to be continuous lattice,then we get the representation of the continuous lattices.To further strengthen the condition,we can get the representation of algebraic lattices,which is different from the algebraic topped intersection structures.For completely distributive lattices,using the condition and results of Deng's,we get the representation of completely distributive lattices.As everyone knows,the open sets of a topological space are a special family of subsets,we get the conclusion that all the open sets form a completely distributive lattice if and only if it satisfies the mentioned condition given by us.And on the basis of Xu's work,further result about the Scott topology on a continuous poset is obtained.As a generalization of O-convergence,02-convergence is also an important con-vergence on posets.Zhao and Li studied on this convergence,and some conditions of the O2-convergence to be topological are obtained.We further investigate the conver-gence.By defining a special relations on posets,we propose the necessary and sufficient conditions for the O2-convergence to be topological.Whether FS-domain and RB-domain are equivalent puzzled almost all the schol-ars in the field of Domain theory.In this paper,using the definition of strong finitely separation functions which we introduce,we obtain a new domain between FS-domain and RB-domain,called it SFS-domain.We prove that in the case of L-domains,or con-sistent meet semilattices or consistent join semilattices,FS-domain and SFS-domain are the same.And by constructing a series of functions,we prove that SFS-domain and RB-domain always equal,that is to say,we give an equivalent characterization of RB-domain.Therefore,in the case of L-domains,or consistent meet semilattices or con-sistent join semilattices,FS-domain and RB-domain are the same.Finally,we obtain a necessary and sufficient condition for an FS-domain to be a RB-domain.