Some Properties Of Strong Raney Posets And Hc-posets

Domain theory appeared in the early part of 1970s when D. Scott was led by prob-lems of semantics for computer languages to initiated the study of continuous lattices. Atabout the same time, Lawson, Stralka, etc. defined a kind of complete lattices with spe-cial properties to characterize a class of compact semilattices. They soon recognized thatthe complete lattices they defined were precisely the continuous lattices given by Scott.Since then,research on continuous lattices and more general the structure of lattices or-dered with some kind of continuity gradually becomes an important direction concernedby mathematics and theoretical computer science community.Completely distributive lattices and continuous lattices have many similarities,interms of subset system,the former are attached to subset system formed by all the sub-sets,and the later attached to a directed subset system. As early as 1952, Raney givessome properties of complete distributive lattices in his several classic paper, and provedthat complete distributive lattice L is a complete distributive lattice if and only if anydifferent point x, y in L can be separated by a main filter'complementary set and a princi-pal ideal'complementary set. since the characterization does not use intersect and unionoperation, only involving order relations, then naturally you can discuss correspondingcharacter on any more general partially ordered set.Follow Marcel Erne′,Menon and others'ideas,we can consider using several kindsof special subset to separate points in partially ordered set,introducing the concepts ofRaney poset,HC?partially ordered set,HA?partially ordered set,and discuss relatedsome properties of posets. Get that directed complete poset P is a strong Raney poset ifand only if P is not only Raney poset but also the A-partially ordered set,Demonstratedstrong Raney poset under the full mapping which has a upper adjoint and a lower adjointis still strong Raney poset.On HC?partially ordered sets,HA?posets we also have theCorresponding conclusions.