A Representation Of Arithmetic Semilattices

Domain theory mainly studies some posets with special properties.Its purpose is to provide a mathematical model for the denotational semantics of programming languages.Because of its obvious background in computer science and its research involving topological and ordering structures,this theory has attracted the common attention of many scholars in theoretical computer science and mathematics.Moreover,many scholars attempt to apply Domain theory to other fields of mathematics.the interaction and mutual induction between topological structure and ordered structure is one of the main characteristics of Domain theory.Among them,searching for special topological structure which can describe some special domain structure equally is a research hotspot of Domain theory.However,the condition of classical topological structure is too strong,and there are some limitations in the representation of domain structure.It is noticed that closed system are generalizations of classical topological structures.Many scholars try to use closed system to represent some domain structures.In this paper,by generalizing classical closure systems,we come true the representation of arithmetic semilattices from the perspective of category equivalence.This master's thesis is divided into two chapters:Chapter 1,we recall some basic concepts and related properties of this thesis,and presents the historical background.Chapter 2,Firstly,we introduce the concept of CX-space,and give some properties of CX-space.Secondly,we give the concept of CX-morphism,and show the one-to-one correspondence between CX-morphisms and Scott continuous maps.Finally,we prove the category which consists of CX-spaces and CX-morphisms and the category which consists of arithmetic semilattices and Scott continuous maps are categorically equivalent.