Some Studies On Consistently Semicontinuous Dcpo

In this paper,we mainly introduce the concept of consistently semicontinuous Dcpo and discuss some properties of it.Firstly,we study the properties of the compatible semicontinuous lattices,and then discuss the semicontinuous Dcpo.The main contents of this paper are as follows:The first part: the concept of semicontinuous lattices is given,and some properties of basis and local basis of semicontinuous lattices are discussed,then equivalent characterizations among basis,local basis and consistently ? are given.The second part: the concept of consistently semicontinuous Dcpo is given,and then powerful consistently semicontinuous Dcpo and consistently semicontinuous mapping are given on the base of consistently semicontinuous Dcpo.And the connection between powerful consistently semicontinuous Dcpo and consistently semicontinuous Dcpo are built by means of consistently semicontinuous mapping and Galois connection.Finally,the concept of basis and local basis on consistently semicontinuous Dcpo is given,then some properties of them are discussed.The third part: we use the existing theory to define the consistently semicontinuous Scott topology,consistently semicontinuous Lawson topology,and then some properties are discussed.Finally,order-preserving mapping on consistently complete lattices is discussed.