Research On Quotient Structures And Congruence Relations Of Domains

Domain theory is a denotational semantic model of computer functional programming languages,objects of study are some special posets,such as directed complete posets,continuous directed complete posets,continuous semilattices and L-domains.Therefore,it can be seen as a branch of order structures.In this paper,we focus on congruence relations and quotient structures of domains.The details are as follows:First,we discover and revise the errors about the part of "homomorphisms and congruence relations of L-domains" in the monograph《Continuous Lattices and Domain Theory》.The concept of strong homomorphisms among L-domains is introduced.We prove that the image of an L-domain under a strong homomorphism is still an L-domain.Base on strong homomorphisms of L-domains,we define the concept of I-congruence relations on L-domains.Meanwhile,we verify that the quotient structure induced by the congruence relation on L-domains with a suitable order is an L-domain in such a way that the quotient map is a strong homomorphism of L-domains.Furthermore,we get the Isomorphism Theorem for L-domains.Second,we investigate the quotient structure of domains.We introduce the concept of congruence relations on domains.We obtain that there is a bijection from the set of all closure operators of the domain P which preserve directed sups onto the set of all congruence relations of P which exclude P×P.Moreover,we propose a new homomorphism between two domains called a strong Scott continuous map.We conclude that kernels of strong Scott continuous maps from a domain and congruence relations on it are practically the same thing.Therefore,we get the homomorphism and Isomorphism theorems for domains.Besides,applying the conclusions about quotient structures of domains to continuous semilattices,we give a positive answer to an open problem on homomorphisms and quotients of continuous semilattices in the monograph《Continuous Lattices and Domain Theory》.We know that coequalizers in the category of topological spaces exist and they correspond to quotient spaces.The coequalizers in arbitrary reflective subcategory of the category of topological spaces also exist.Thus,coequalizers could be used to study quotient structures.Note that the category of sober spaces is an important reflective subcategory in the category of T0 spaces.For studying the quotient structure of the domains better,we need to investigate into sober spaces deeply.At last,we explore sober spaces and their related structures from different perspectives.We extend the descriptive set theory of second countable sober spaces to first countable sober spaces.Next,we propose a counterexample to explain that if Y is sober,the function space TOP(X,Y)equipped with the Isbell topology(respectively,Scott topology)may be a non-sober space.Furthermore,we provide a uniform construction to d-spaces and well-filtered spaces by introducing the concept of H-well-filtered spaces.We obtain that for a T0 space X and an H-well-filtered space Y,the function space TOP(X,Y)equipped with the Isbell topology is H-well-filtered.Beyond the aforementioned work,we solve several open problems concerning strong d-spaces posed by Xiaoquan Xu and Dongsheng Zhao.