Researches On Category Duality Of Posets And Topological Properties

Domain theory was proposed by D.S.Scott in the early 1970s,it provided the denotational semantics of programming languages.Order structure and topolog-ical structure are important mathematical structures in domain theory,they can be generated by each other.In this paper,on the basis of the existing domain the-ory and the latest research results,we discuss the topological representation and category duality for a new class of posets;give the category duality theorem about Z-predistributive or Z-precontinuous posets;study some non-Hausdorff topologi-cal properties.The details are as follows:First,we define a new class of posets which are complemented and ideal-distributive,we call these strong Boolean posets.This definition is a generalization of Boolean lattice on posets,and they are different from the Boolean posets.We give a topology on the set of all prime Frink ideals on the strong Boolean posets in order to obtain the Stone’s topological representation for strong Boolean posets.A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented.Second,we research further into ZZ-predistributive and Z-precontinuous posets introduced by Erne.We focus on duality theorems based on the application of Galois connections whenever Z is a closed subset selection.For example,there is a duality between the categories Z-PDG and Z-PDD of all Z-predistributive posets with weakly Z△-continuous maps which have a lower adjoint,and the maps preserve Z-below relations that have an upper adjoint,respectively,as morphisms.We introduce the concept of Z0-approximating auxiliary relation,and have made a slight improvement on Z-precontinuity,so that there is a generalization of the classical equivalence between domain and auxiliary relation.Third,sobriety,well-filteredness and monotone convergence are three of the most important properties of topological spaces extensively studied in domain the-ory.Some other weak forms of sobriety and well-filteredness have also been investi-gated by some authors.The concepts of these topological properties look different.We introduce the notion of(?)-fine space,which provides a unified approach to such properties.In addition,we also do further research on weak well-filtered and weak sober spaces.We prove that weak well-filteredness and weak sobriety are equiva-lent in locally compact spaces.In the first countable spaces,weak sobriety is also equivalent to weak well-filteredness.Finally,we continue to investigate the(?)-fine spaces mentioned above.More-over,this unified method also puts forward some new topological properties:PF-sobriety and PF-well-filteredness.We explore PF-sober and PF-well-filtered s-paces,and find that they are strictly weaker than weak sobriety and weak well-filteredness respectively,and they are different from other spaces in terms of co-herence or retraction.