Relative Filter On Poset And Its Decomposition

Domain theory is an intersection between the theoretical computer science and mathematics.Since 1970s,Domain theory was always being focused by computer science and mathematics.As a research object,special elements ideals and filters attract much attention.In this thesis,we mainly study relative filter and prime strong ideal,which used as the generalization of ideals and filters on posets.The main work summed up in the following aspects:In the second chapter,we introduce and examine the relative filtered sets on posets.In the third chapter,we give the notion of the relative filter and the locally maximal relative filter on posets,and prove the existence of the locally maximal relative filter.We also explore the relationship among the filter,the relative filter and uniform filter.In addition,we obtain that every relative filter of posets can be decomposed as intersection of some locally maximal relative filters.In the fourth chapter,we investigate some basic properties of the relative du-al way below relation and the relative dually continuous posets,and obtain some intrisinc characterizations of them.In the fifth chapter,we study some applications of the strong ideal on the poset.These theorems generalize corresponding results in reference[29]from finite posets to posets.In addition,the concept of prime strong ideal on posets is introduced and examined.We conclude that the image of a strong ideal under a homomorphic mapping is still a strong ideal and the image of a prime strong ideal under an injective homomorphic mapping is still a prime strong ideal.