Galois Connections On Metric Spaces

Metric space is an important tool to understand topological spaces and order structures.A Galois connection is a pair of special mappings between posets and plays an important role in studying ordered structures.In recent years,as a generalization of Galois connection,the notion of adjoint functor plays an important role in the study of tensored metric spaces.On the other hand,topological spaces and ordered structures are deeply related,which is most obvious in Domain theory.Recent studies show that many special properties of metric spaces can be described by means of ordered structures.In this paper,we try to add partial order structures into metric spaces and discuss the categorical properties of the special ordered metric structures with the notion of Galois connection as a tool.Firstly,we introduce the concept of metric semilattice and an operation on such structures,and then prove that this operation can serve as a tensor product on metric spaces.Secondly,by introducing the concept of cm-semilattices and tcm-semilattices,we obtain the pseudo embedding between cm-semilattice,and study the properties of TCMsep+ which composes of tcm-semilattices and tensor preserving M-mappinsg.Finally,as a special case of metric semilattices,the concept of complete cm-lattices is introduced,and the detailed properties of complete cm-lattice(LI,d,?)induced by any given complete cm-lattice(L,d,?)and set I are presented.