Quasi-hypercontinuous Domains And Quasi-hypercontinuous Lattices

Domain theory appeared in the early part of 1970s when D. Scott was led by problems of semantics for computer languages to initiated the study of continuous lattices. At about the same time, in the field of pure mathematics Lawson, Stralka, etc. defined a kind of complete lattices with special properties to characterize a class of compact semilattices. They soon recognized that the complete lattices they defined were precisely the continuous lattices given by Scott. As the common objects studied in theoretical computer science and pure mathematics, continuous lattices attract considerable attention and a great many results about continuous lattices were obtained. In 1980, Gierz, etc. wrote the monograph " A compendium of continuous lattices " to summarize the work, which symbolized the maturity of theory of continuous lattices appearing as the early form of domain theory. The complementary version " Continuous Lattices and Domains " absorbing the results of the latest twenty years was published in 2003.The main objects in domain theory are partially ordered sets (posets for short) and the mappings between posets. One of important aspects of domain theory is to carry as much of the theory of continuous lattices to general ordered structure as possible. As one of the most successful generalizations of continuous lattices, Gierz and Lawson introduced the concept of generalized continuous lattice (renamed to quasicontinous lattice in [22]) in [23]. A complete lattice L is a quasicontinuous lattice iff the lattice of all Scott open subsets of L is hypercontinuous and hypercontinuous lattices are continuous lattices which are Hausdorff in their interval topology. As the common generalization of quasicontinuous lattice and continuous domain, Gierz, Lawson and Stralka introduced an important kind of domain, quasicontinous domain. From the point of view of...