Several Properties On The Z-subset System

In this thesis, Some properties of Z - subset systems on posets are discussed, including some Properties of Z -Scott Topology, Z -Lawson Topology and Z - Scott Open filters.In the first chapter, the background and relevant progress of the thesis is given in the preface. The basic concepts of partial posets,and some basic results used in the thesis is given in propaedeutics.Second chapter that is divided into four sections. Z-subset system and Z-continuous partial order is introduced in the first section. The definition of Z -way-below relation is discussed, The Z-distribution law on the Z-continuous posets, and some other properties. In sec-tion 2, Z-Scott topologies, Z-Lawson topologies, and double Z-Scott topologies on an ordered set, Some properties of Z-Scott topologies and double Z-Scott topologies are discussed.Partial ordering is discussed the third chapter, Z - Scott topology,the Ti separation property of the Z - Lawson topology continuous posets and the Z - Scott extension is discussed on the basis of Z - quasi-continuous posets, And the Z - Lawson Topology T2 separability is discussed, In Section 4, the definition of Z-Scott filter is given, and U is a Z-Scott open, The necessary and sufficient conditions for the filter are defined. The way -below relation between the sets is defined and the set, There are some relationships between the way-below and Z-Scott open filters.In chapter 3, Z - continuous mapping, double Z - continuous map-ping and Z - Scott is given firstly, Continuous Z - Scott continuous mapping and the double Z - Scott continuous mapping are discussed,And Z - continuous mapping and double Z - Scott continuous map-ping, The sufficient and necessary conditions for the double Z - Scott continuous mapping are obtained.