Some Studies On Relative Continuous Posets Theory

Since its birth,Domain theory,with the theoretical background of computer programming language,has attracted a fervent attention in various sciences,especially in the field of intersection of mathematics and computer science.Nowadays,Domain theory has become an irreplaceable branch of mathematics.Domain theory will be enriched and perfected greatly,if the directed set,as one of the most basic and important definitions of Domain theory,can be further generalized.In this paper,the relatively directed set and some new definitions are introduced on the posets by the ideas of relativity.Furthermore,the relatively continuous poset is presented and some its properties are researched by some related concepts.The main research work is as follows:1.On the basis of the concepts of directed set and uniform set,the relatively directed set and relative directed complete set are introduced and examined.It will be obtained that the family of all directed sets relative to the directed set T on posets is a complete lattice.2.Relative ideal and relative maximal ideal are presented,and their respective properties are discussed.Moreover,the existence of relative maximal ideal will be proved and clarify the relationship among ideal,relative ideal and consistent ideal will be studied.3.By using the relative way below relation,the relatively continuous poset is defined and some its equivalent characterizations will be given.Also,it will be proved that the relative continuous posets has the heritability of relative to T under a given set T.4.The relative basis and weight of the relatively continuous Domain are intro-duced and its some equivalent characterizations and basic properties are given.5.Relative Scott topology is defined by using the relative neighborhood and the relative Scott open set,and the insertion property of relative way below relation of relative continuous posets in the given set T is proved.