Representing Domains As Information Systems And Study Of Some Generalizations Of Domains

Domain theory is one of the important research fields of theoretical computer science. Mutual transformations and infiltration of order, topology, approximation and logic are the basic features of this theory. With the continuous development of its own, domain theory interacts with artificial intelligence and information science, and reflects certain background of applications. So far, various notions of generalized domains have been introduced and a large volume of work on generalized domains has been done. These generalizations not only enrich domain theory but also have their own research perspectives.In this paper, we are mainly concerned with the following two aspects. For the first aspect, we investigate (algebraic) domains via information systems, a logic-oriented ap-proach. The second chapter is some of the latest work in this area. For the second aspect, we explore further several generalizations of domains and several current problems related to them. The third chapter to the fifth chapter is the corresponding work.In the second chapter, within continuous information systems (C-inf), concepts of algebraic information systems (A-inf), weak algebraic information systems (wA-inf) are introduced. Properties and relationships between them are explored. Based on these, some new work on representing (resp., algebraic) domains as information systems is done. Main results are:(1) A dcpo D is a domain (resp., algebraic domain) iff there is a representation of D via a C-inf (resp., an A-inf); (2) every A-inf is a GA-inf and a wA-inf; (3) every C-inf which represents an algebraic domain is a GA-inf, and every GA-inf represents an algebraic domain; (4) every induced C-inf S(D, B) of a domain is a wA-inf; (5) the induced domain of a continuous B-information system (cB-inf) is a continuous B-domain; (6) a domain D is a BF-domain iff there is a representation of D via a BF-inf.In the third chapter, in terms of principal ideals, principal filters and closed intervals, the quasicontinuity of a dcpo is investigated. Some sufficient conditions for a dcpo to be quasicontinuous are given. Besides, some new properties of quasicontinuous domains are obtained. It should be pointed out that, different from the case for continuity of a dcpo, one cannot deduce the quasicontinuity of a dcpo by checking every its principal ideal being quasicontinuous. However, we proved that a dcpo having finitely many maximal elements is quasicontinuous iff every principal ideal is quasicontinuous. Similarly, the C-continuity of posets are studied. The concept of principal ideal C-continuity for posets is introduced. By lifting and principal ideal C-continuity, two new characterizations for C-continuous posets are given. Moreover, the concept of QC-continuous posets, a generalization of C-continuous posets is introduced. It is proved that a complete lattice L is a generalized completely distributive (GCD) lattice iff it is quasicontinuous and QC-continuous. In terms of QC-continuity, it is proved that quasicontinuity of a poset is equivalent to the quasicontinuity of the lattice of its Scott-closed sets. So, a poset L is quasicontinuous iff the lattice of its Scott-closed sets σ*(L) is a GCD lattice iff σ-*(L) is a quasicontinuous lattice iff the Scott topology σ(L) is a hypercontinuous lattice. It is obtained that σ*(L) of a dcpo L satisfying the property M is C-algebraic. A new sufficient condition is given for the isomorphism of two dcpos with isomorphic lattices of Scott-closed sets.In the fourth chapter, we are concerned with the closedness of the class of QFS-domains by taking lifts, the Hoare powerdomain, the Smyth powerdomain and Scott-continuous projections. We obtain that:(1) A poset L is quasicontinuous iff the lattice of Scott-closed sets σ*(L) is a QFS-domain; (2) images of QFS-domains under Scott-continuous projections are QFS-domains; (3) the Hoare powerdomain H(L) of a QFS-domain L is a QFS-domain; (4) A dcpo L is quasicontinuous iff the Hoare powerdomain H(L) of L is a quasicontinuous domain; (5) the Smyth powerdomain QL of a QFS-domain L is an FS-domain, particularly a QFS-domain.In the fifth chapter, we firstly investigate the continuity of the distributive reflection of a semicontinuous lattice. We prove that for a complete lattice L, the distributive re-flection Ld is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. We construct counterexamples to solve two open problems posed by D. Zhao. Then the concept of strongly semicontinuous lattices, a new class of complete lat-tices lying between the class of semicontinuous lattices and that of continuous lattices, are introduced and studied. It is shown that a complete lattice L is strongly semicontinuous iff L is semicontinuous and meet semicontinuous. Some versions of strong interpolation prop-erties for the semi-way-below relation in (strongly) semicontinuous lattices are established. Characterization theorems by some distributivity are given. Furthermore, the concepts of strongly semicontinuous domains and semi-FS domains are introduced. It is proved that semi-FS domains are strongly semicontinuous, and strongly semicontinuous bc-domains, in particular strongly semicontinuous lattices, are all semi-FS domains.