Some Studies On The Theory Of Consistent Join-semicontinuous Lattices

In 1971, Scott D put forward the concept of continuous lattices in the first time, the classical theory of Domain began to emerge. In 1989, Ray Y firstly proposed the concept of semiprime ideal and some basic properties of it were investigated. For the first time in 1997, Zhao Dongsheng took advantage of semiprime ideal, giving a new relationship between point and point on complete lattices, and then proposed the concept of semicontinuous lattices. And then, the theory of semicontinuous lattice has been developing rapidly. Inspired by this, we put forward the concept of consistent semiprime filtes by way of semiprime filtes, then we proposed the concept of consistent join-complete lattices making use of consistent semiprime filtes, further we investigate a relationship which having a role in the relationship between point and point on the basis of consistent join-complete lattices, and then we obtain the definition of consistent join-semicontinuous lattices and do researches on some properties of them. In addition, some mapping properties of consistent joinsemicontinuous lattices are discussed. Finally, we promote the relationship “?(8”between point and point to set and set so that to give the definition of quasi-consistent join-semicontinuous lattices. The main contents are as follows:The first part: We introduce two kinds of new relationships“?”and“?(8”so that to come up with the concept of consistent semiprime filtes, further put forward the concept of consistent join-complete lattices, and then do researches for some properties of them. On this basis, the concept of consistent join-semicontinuous lattices is introduced, and do research on some properties of them.The second part: We introduce the concept of mapping of consistent joinsemicontinuous lattices, some mapping properties of consistent join-semicontinuous lattices are discussed, the fact that any retract of a consistent join-semicontinuous lattice is still a consistent join-semicontinuous lattice is investigated.The third part: We promote the relationship “?(8”between point and point to set and set so that to give the definition of quasi-consistent join-semicontinuous lattices, and some basic properties of it are investigated.