The Research On Induced L-Convex Spaces And Related Theory

This paper mainly includes two parts.Part I is Chapter 3.In this part,the properties of induced L-convex spaces are studied.Part II is Chapter 4.In this part,convexity-preserving mappings between M-fuzzifying convergence spaces and closure-preserving mappings between M-fuzzifying preconvex closure spaces are discussed.In the following,the main work is explained in detail.Chapter 1 is the introduction to understand the process of research work.Chapter 2 mainly gives the relevant concepts and conclusions of lattice theory and lattice valued convex structure theory.In Chapter 3,firstly,the relation between induced L-convex spaces and L-hull operators is studied.Secondly,we discuss the relationship of induced L-convex spaces with product spaces.It is shown that the induced L-convex structure by the product convex structure contains the product L-convex structure of the induced L-convex structure.Thirdly,the relationship of induced L-convex spaces with quotient spaces is studied.It proves that the quotient L-convex structure of the induced L-convex structure is exactly the induced L-convex structure by quotient convex structure.Finally,we introduce sub-S1,sub-S2,S2 and sub-S3 separation axioms in L-convex spaces,and investigate their hereditary property,productive property.The relationships of L-CC mapping,L-CP mapping between a convex space and the induced L-convex space are discussed.Chapter 4 proposes the lattice-valued forms of convexity-preserving mappings between M-fuzzifying convergence spaces,and closure-preserving mappings between M-fuzzifying preconvex closure spaces.We discuss the relationships of M-fuzzifying convexity-preserving mappings with M-CP mappings and M-fuzzifying preconvex closure operators in M-fuzzifying convergence spaces.Moreover,we discuss the relationships between M-fuzzifying convexity-preserving mappings and separation properties in M-fuzzifying convergence spaces.In Chapter 5,conclusion remarks and expectation are made.