Semiprime Minimal Sets Of Semicontinous Lattice And Extension Of Homomorphisms

In the recent thirty years, the interactions of computer science and mathematics, especiallythe applications of topological structures, lattice-ordered structures and category structures incomputer science, have attracted a considerable deal of attention. In the 1970,s , Scott, Plotkin,Lawson and Hofmann established the theory of continuous lattices(domains), and the structure oftheory 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 as-pects 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, and 1990,, hypercontinuous lattices,generalized continuous lattices, Z-continuous posets and FS-lattices were introduced by Gierz, Law-son, Keimel, which are among the most successful generalizations of continuous lattices(domains).In 1983, 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 pointsto the case of sets.One main aim of this paper is to investigate some properties, consequently, the concept ofsemiprime minimal sets is introduced for complete lattices. It is proved that a complete lattice Lis semicontinuous if and only if every element a of L has a semiprime minimal set. Two extensiontheorems of homomorphisms between semicontinuous lattices are obtained.Another aim of this paper is to investigate the mapping properties of Z-semicontinuous lat-tices.