Pseudo-Hypercontinuous Poset

Domain theory appeared in the early part of 1970s when Scott was led by problems of semanticsfor computer languages to initiated the study of continuous lattices. At about the same time, Lawson,Stralka, etc. defined a kind of complete lattices with special properties to characterize a class of compactsemilattices. They soon recognized that the complete lattices they defined were precisely the continuous lattices given by Scott. Since then,research on continuous lattices and more general the structure of latticesordered with some kind of continuity gradually becomes an important direction concerned by mathematicsand theoretical computer science community.Completely distributive lattices are important distributive lattices, Raney and other people have make deep research on them . In terms of subset system, completely distributive lattices and continuous lattices have similarities, the former are attached to subset system formed by all the subsets, and the later attached to a directed subset system.As early as 1953, Raney gave the characterization of complete distributive lattices , and proved that complete distributive lattice L is a complete distributive lattice if and only if any dierent point x; y in L can be separated by a main filter'complementary set and a principal ideal'complementary set. since the characterization does not use intersect and union operation, only involving order relations, this means it only involves the characterization of order relations of L,then naturally you can discuss corresponding character on any more general partially ordered set.Follow Marcel Erné,Menon and others'ideas,we can consider using several kinds of special subset to separate points in partially ordered set,introducing the concepts of pseudo- hypercontinuous poset, pseudo-hyperalgebraic poset ,and discuss their basic properties. Demonstrated pseudo-hypercontinuous poset under the full mapping which has a upper adjoint and a lower adjoint is still pseudo-hypercontinuous poset.On pseudo-hyperalgebraic poset, we also have the Corresponding conclusion.