The Representations Of Arithmetic Semilattice By Information Systems And Closure Spaces

In the 1970's,Domain theory was proposed by Dana Scott,A Turing Award recipient.This theory provides denotational semantics for functional programming languages and plays an important role in Theoretical Computer Science.Domain the-ory combines lattice theory,category theory,topology and computer science;therefore,not only has it been attracting attention from computer scientists,but also becomes an important branch of mathematics research.During the past decades,although Domain theory has been developed marvelously,there are still problems worth exploringIn the research of various substructures in Domain theory,scholars have estab-lished different representation theories for those structures,such as topological char-acterization,closure characterization,formal concept analysis representation and in-formation system representation.In 1982,Scott proposed the concept of information systems and approximate mapping as a logical approach to denotational semantics of programming languages.Later,Hoofman gave the information system representation of bounded complete continuous domain.At the same time,Vickers used the theory of information system to represent continuous posets and also proved that his infor-mation system can represent all continuous domains,despite the fact that his system not being in the Scott-style.Until 2008,Spreen and Luoshan Xu introduced the con-cept of general continuous information system of Scott continuous dcpos.In addition,they obtained that the general algebraic information system can be used to represen-t algebraic domains.In 2014,Mingyuan Wu and Qingguo Li proposed the concept of algebraic information system with more concise conditions and obtained that their algebraic information system represent algebraic domains as wellBased on the research of algebraic information system mentioned above,this thesis presents a new information system by adding appropriate conditions.We prove that our new information systems represent arithmetic semilattices.In this paper,The first concept is arithmetic information system which proves out to be the information system representation of arithmetic semilattices.From a categorical viewpoint,we proved that the category of arithmetic information systems with approximate mappings as morphisms is equivalent to that of arithmetic semilattices with continuous functions as morphisms.The representation of order structure by family of subsets have always been an interesting topic in mathematics.In 1930s,Stone represented the Boolean lattices by certain family of sets in Stone spaces.Later,Priestley generalized this result and ob-tained the representation of bounded distributive lattices.In 2015,Lankun Guo and Qingguo Li proposed the notion of F-augmented closure spaces.This work provides a new approach to representing algebraic domains.Based on F-augmented closure s-paces,we give a new kind of closure spaces by adding an appropriate condition and prove that our new spaces are representations of arithmetic semilattices.Besides,we in-troduce the notion of arithmetic approximate mappings which serve as the appropriate morphisms for arithmetic closure spaces.From categorical perspective,we obtain an equivalence between the category of arithmetic closure spaces with arithmetic approxi-mate mappings as morphisms and that of arithmetic semilattices with Scott continuous functions as morphisms.