Quasi-Continuous Compatible Dcpo And Supercontinuous Compatible Dcpo

In thesis, we introduced the concept of quasi-continuous Cdcpo and super-continuous Cdcpo based on quasi-continuous Domain and super-continuous Domain. The main contents are as following:The first part of this thesis, we give some basic notions, notations and results concerned with Cdcpo.In the second part, we first generlized the way-below reletion be-tween points to the relationship between sets, give the definition of quasi-continuous Cdcpo. After generalizing the Rudin lemma, we give the characterization of Q-Cdcpo and QA-Cdcpo. Secondly the definition of compatible quasi-base and compatible local quasi-bases and compatible quasi-minimal sets are given. Then we obtained some characterization of compatible quasi-base and compatible local quasi-bases and compatible quasi-minimal sets. We investigate the preser-vation of Q-Cdcpo under some mappings. We showed that Q-Cdcop is preserved under compatible Scott quasi-continuous mapping and C-compatible continuous mapping. Finally, we prove that Q-Cdcpo are hereditary for the Scott open sets and Scott closed sets, and as for retraction.In the third part, we first define the concept of super-continuous Cdcpo. Some equivalent characterizations and properties are also given. Secondly, it is proved that the finite product of super-continuous Cdcpo is also super-continuous Cdcpo. Then we prove super-continuous Cdcpo’s retraction is still super-continuous Cdcpo and is hereditary for its sub-compatible Dcpo. At last, we investigate the relation-ship among super-continuous Cdcpo, meet-continuous Cdcpo quasi-continuous Cdcpo.