Some Properties Of Exact Semicontinuous 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. The main objects in domain theory are partially ordered sets and the mappings between posets. The main contents of the thesis consists of two parts. Firstly, we give some characterizations of several kinds of posets by using separating of points. Secondly, we introduce the concept of exact semicontinuous lattices and discuss some properties of exact semicontinuous lattices.Raney gave an intrinsic characterization of completely distributive lattices by only using order relation in his classical papers. Since this characterization does not depend on the meets and joins, one can look at arbitrary posets with this property. Then some other experts have studied several kinds of posets that have this similar characterization. Following these idals, we collect some partially ordered sets that can separate points with special subset, and we prove that for any distinct two points in countably approximating posets or semicontinuous lattices or semialgebra lattices there exist an upper set and a lower set that separate the two points.In 2007, in order to study the models of topological spaces Joe Mashuburn introduced the concept of exact posets and proved that every first countable space is homeomorphic to the maximal points space of a weakly domain, where a weakly domain is an exact poset with weakly increasing auxiliary relation. We introduce the concepts of exact semicontinuous lattices and discuss some properties of them.