The Research Of Order-convergence In Posets Being Topological

In the 70s of last century,theoretical computer scientist Scott proposed the Domain Theory from the perspective of computer science.Since Domain Theory is established on the fusion of order and topology structures,it has attracted great attention of re-searchers in mathematics.In the following 40 years,one of the important research topic is to study the Domain structure by using the Convergence Theory.In this respect,Scott introduced the notions of S-convergence in Dcpo and the Scott topology,and proved that the equivalence between the S-convergence being topological and the continuity of a given Dcpo.This indeed can be viewed as an equivalent characterization for Domain.Based on the ideal of Scott,in this paper,we first define the MN-convergence as the unified form of the O-convergence and the O2-convergence in posets.And then,we explore the characterizations for the O1-convergence,the MN-convergence and the lim-infM-convergence being topological,respectively.In chapter two,inspired by the way-below relation<<in Domain Theory,we pro-pose two new approximate relations<<s and(?)s on posets in which<<s is strictly weak than<<.On one hand,replying on the definition of approximate relations<<s and(?)s,we introduce the notion of B-consistent S*-doubly continuous posets.On the other hand,we study the B-topology which is determined by O1-convergence according to the standard topological approach.By establishing the characterization theorem for the B-topology,we obtain a sufficient and necessary condition for the O1-convergence to be topological.In chapter three,since the concept of B-consistent S*-doubly continuous posets seems to be abstract,by using some concepts which originally come from Lattice Theory and Set Theory,two special sub-classes of B-consistent S*-doubly continuous posets,named locally ordinal posets and locally complete posets,are defined and investigated.In chapter four,by introducing the double-systems of sets for a given poset,we first proposed the concept of MN-convergence in posets as the unified form of the O-convergence and the O2-convergence.Then,by studying the fundamental properties of MN-topology which is determined by MN-convergence according to the standard topological approach,an equivalent characterization theorem for the MN-topology is established.By this theorem,an characterization for the MN-convergence being topo-logical is obtained.In chapter five,by introducing system of sets for a given poset,Zhou defined the notion of lim-infM-convergence as the unified form of the lim-inf-convergence and the lim-inf2-convergence in posets.We continue to explore the characterization for the lim-infM-convergence being topological.By establishing a equivalent description for the ap-proximate relation<<α(M)which was originally defined by Zhou,we propose the notion of α*(M)-continuous posets.It is worth noting that an α*(M)-continuous poset may not be an α(M)-continuous poset but it has a closed relation with α(M)-continuous poset.Also,the MN-topology on posets is defined and some fundamental properties of it are studied.Finally,we prove the equivalence between the lim-infM-convergence being topological and the α*(M)-continuity of a given poset.