| In the recent thirty years, the interactions of computer science and mathematics, especially the applications of topological structures, lattice-ordered structures and category structures in computer science, have attracted a considerable deal of attention. In the 1970's, Scott, Plotkin, Hofmann, Lawson and Keimel established the theory of continuous lattices (domains), and the structure theory of domains has been a focal point for research in denotational semantics ever since.From both the computer science side and the purely mathematical side, one of important aspects of domain theory is to carry as much as possible of the theory of continuous lattices (domains) to as general an ordered structure as possible. In the 1980's and 1990's, hypercontinuous lattices, generalized continuous lattices, Z-continuous lattices (posets) and FS-lattices were introduced by Gierz, Lawson, Bandelt, Erne, Huth, Jung and Keimel, which are among the most successful generalizations of continuous lattices (domains). In 1983, as a common generalization of continuous domains and generalized continuous lattices, an important class of domains (called quasicontinuous domains) was introduced by Gierz, Lawson and Stralka. Their basic idea is to generalize the way below relation between the points to the case of sets.One main aim of this paper is to generalize the quasicontinuous domain theory to a general subset system Z. We will do it in two different ways.One way to generalize the quasicontinuous domain theory is to follow the idea of Gierz, Lawson and Stralka and the Rudin lemma. We first generalize the Rudin property to a general subset system Z and give some characterizations in mapping forms. They provide a basis for generalizing quasicontinuous domains to a generalsubset system Z. As a common generalization of quasicontonuous posets and Z-continuous posets, quasi Z-continous domains are introduced in the paper. It is proved that if Z satisfies certain conditions, then the Z-below relation < |
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