The Categorical Equivalence Between Closure-based Information Systems And Continuous Domains

The representation of domain structures is a hot research direction of domain theory.The purpose is to represent abstract domain structures by relatively concrete or simple mathematical structures.The common tools for this topic are closure spaces and Scott-type information systems.Recent study shows that formal concept analysis provides a new approach for representing domain structures.This paper proposes a new type of information system-s by combining closure spaces and Scott-type information systems with the aim of representing continuous domains.Firstly,we study the properties of Scott-type information systems induced by relationally consistent formal con-texts,and propose the notion of closure-based information systems.Then,we study the relationship between closure-based information systems,relational-ly consistent formal contexts and continuous domains.Specifically,using the concept of information states,we prove that every closure-based information system can generate a continuous domain;and conversely,every continuous domain is a partial ordered set,in the sense of order isomorphism,composed of all information states of a closure-based information system.Finally,we establish the category equivalence between closure-based information systems and continuous domains.The thesis is constructed as follows:In Chapter 1,the related research background is stated,and some basic concepts and related properties are given.In Chapter 2,based on the research of information system induced by relation-ally consistent formal context,the closure-based information system is intro-duced,and the internal relationship between the two structures is studied.In Chapter 3,this paper introduces the F-morphism between closure-based infor-mation system,and studies the internal relationship between the F-morphism and the F-approximation mapping between relationally consistent formal con-text.In Chapter 4,it is proved that the category of closure-based information system and F-morphism is equivalent to the category of continuous domain and Scott continuous mapping.