Research On Several Problems Of Z-Continuous Posets Theory

Domain theory produced in the early 1970s is an important area of theoretical computer science, designed for the study of computer functional languages to lay the foundation of mathematics. The interaction between order and topology plays an important role in this theory. When continuous lattices theory developed, They have been studied extensively by many people from various fields due to its strong connection to algebra, logics, general topology and computer science. This dissertation will do further study on domain theory, in particular the theory of Z-continuous posets.First of all, some results about domains and Z-continuous posets are sum-marized and supplemented. The properties of the lattice of Z-join ideals are considered, and definitions and facts about Z-algebraic intersection structures, Z-algebraic closure operators and the interrelation between them in the sense of subset systems are discussed. On one hand every Z-algebraic closure operator induces a topped Z-algebraic∩-structure, on the other hand every Z-algebraic∩-structure induced a Z-algebraic closure operator. Moreover, every topped Z-algebraic∩-structure is a Z-algebraic lattice and every Z-algebraic lattice is iso-morphic to a topped Z-algebraic∩-structure. Furthermore the concepts of quasi-compact elements, quasi-bases and special quasi-bases in continuous lattices are introduced. After investigating characterizations of quasi-compact elements and quasi-bases in continuous lattices, a procedure for generating arithmetic lattices by special quasi-bases is given. If B is a special quasi-base for a continuous lattice L, then the poset of all round ideals of B ordered by the inclusion order is an arithmetic lattice. And then the representations for continuous lattices, algebraic lattices and arithmetic lattices are formulated by certain family of subsets of a given set.Second, in combination with the topological theory, on one hand, we proved that the topology on P induced by the class of lim-inf-convergence is exactly the Scott topology on P and that for a poset P the lim-inf-convergence is topolog-ical for the Scott topology if and only if P is a continuous poset. Moreover the meet continuity of general posets is characterized in terms of lim-inf-convergence. Based on the interrelation between lim-inf-convergence and Scott topology, we consider the topological properties of meet continuous posets, enriching the in-terplay between topological and order theoretical aspects of meet continuity of posets. On the other hand, we investigate the problem of extensions of continuous mappings from order theoretical aspects, by means of the family of closed sets as a complete lattice, which is different from just using the topological theory to study the problem in the past. Moreover, a necessary and sufficient condition is provided in order that a continuous mapping from the dense subspace to a T3 space has a continuous extension over the whole space.Again, in combination with the algebraic theory, we consider the relation-ship between the decompositions of the elements in posets and the Z-algebraic. After introducing Z-finiteness and Z-additivity, we study the decompositions, finite decompositions and uniqueness of finite decompositions in complete lat tices. Furthermore, in virtue of some Z-join ideals lattice, we characterize the decompositions in a poset P and prove that if P is P-algebraic and every princi-pal ideals is $mathcalP-finite in the Z-join ideals lattice ZˇP, then P has finite decompositions in Kz(P).Finally, in combination with the category theory, we introduce bifinite upper bounded posets, we also investigate their properties and the DΔ-continuous func-tions between them. For bifinite upper bounded posets R and S, their product and the function space [R→S]D which is the set of all DΔ-continuous functions from R to S ordered by the pointwise partial ordering are also bifinite upper bounded posets. Hence the category BFBP, in which objects are bifinte upper bounded posets and arrows are DΔ-continuous functions between them, is Carte-sian closed.