D - Precontinuous Partial Posets And Semi - Pre - Continuous Posets

The theory of continuous lattices has two aspects of backgrounds of theoretical computer science and mathematics. Since it birth it has been paid close attention. On the one hand, an important task of investigation is to extend continuous lattices to more general ordered structures. On the other hand, because the theory of classical continuous lattices and domains are based on the investigation of dcpos, and it limits the research. So researches that use other collections of subsets to replace directed sets to define new continuities are meaningful.This paper is based on the "classical" subset selection D of all directed sets. The main contents of this dissertation are as listed follows:Chapter Ⅰ. In this part we give a brief introduce about posets, projections, images of projections, semi-continuous lattices and semi-continuous dcpo’s, and then recall the concept of semi-prime ideals and semi-prime sets and related elementary results.Chapter Ⅱ. First, we introduce the cut operator A, and prove some useful properties of it. Second, we prove that the relation of D-way-below <<D on a D-precontinuous poset, has the interpolation property. A characterization theorem for D-precontinuous posets is given.Chapter Ⅲ. First, a counterexample is constructed to show that DΔ-continuous projection images of D-precontinuous posets needn’t be D-precontinuous. Second, we replace projections with kernel operators and prove that images of D-precontinuous posets under DΔ-continuous kernel are still D-precontinuous posets.Chapter Ⅳ. First, we define the new relation "(?)D" and the concept of semi-D-precontinuous lattices in terms of semi-prime ideals. Some properties and characterization theorem s for semi-D-precontinuous lattices are given. Second, in terms of semi-prime sets the relation(?)D is successfully extended to posets and the concept of semi-D-precontinuous posets is defined. Some properties and characterizations for semi-D-precontinuous posets are also given.