Semi-continuous Dcpo

The theory of continuous lattices has two aspects of backgrounds of theoretical computer science and mathematics. So since its birth it has been paid close attention to. One of the important tasks is to extend continuous lattices to more general sequence structures. Y. Rav defined semiprime ideals on lattices and described them. Dongsheng Zhao defined a binary realitionship on complete lattices and the concept of semicontinuous lattices, extending properties of continuous lattices to semicontinuous lattices. Because semicontinuous lattices were defined on the complete lattice, researches on semicontinuous lattices have some limitations. This thesis introduces the concept of semicontinuous dcpo so as to extend the application of semicontinuous lattices, and carries on detailed research. Firstly, equivalent characterizations of prime ideals on poset are proved, and concept of semi-prime ideals on lattice is extended to semi-prime sets on dcpo, then defines the realitionship "(?)" and the concept of semicontinuous dcpo in terms of semi-prime sets. Some properties of semicontinuous lattice will be successfully extended to semicontinuous dcpo. We prove that when L is a complete lattice, semi-prime sets must be semi-prime ideals; and the realitionship ’(?)’" is consistent with Dongsheng Zhao’s. So semi-prime sets and semicontinuous dcpo are reasonable generalizations of semi-prime ideals and semicontinuous lattices. We also prove that every domain is a semicontinuous dcpo, and obtain several equivalent conditions of semicontinuous dcpo.This thesis defines semi-Scott topology and semi-Lawson topology on dcpo"s. And a sufficient condition on semicontinuous dcpo is proved. We prove that an upper set U is semi-Lawson open iff it is semi-Scott open; a lower set U is semi-Lawson closed iff it is semi-Scott closed. The concept of semicontinuous mapping is introduced and then we study their properties. We prove that if f is order preserving and semicontinuous, then f is continuous with respect to the semi-Scott topologies.Finally, the concept of semi-prime ideals on dcpo are introduced, and we prove that prime ideals must be semi-prime ideals; and semicontinuous lattices must be strongly semicontinuous dcpo’s. We also obtain several properties on strongly semicontinuous dcpo.