Studies On Theories Of M-fuzzifying And (L, M)-fuzzy Convex Spaces

In this thesis, we mainly discuss theories of M-fuzzifying convex structures,(L, A/)-fuzzy convex structures and category and subcategories of (L, M)-fuzzy convex structures.In the theory of M-fuzzifying convex structures, we firstly introduce and char-acterize M-fuzzifying (resp. weak) JHC convex structures, M-fuzzifying geometric interval spaces, M-fuzzifying (resp. weak) Peano, Pasch and Sand-glass) interval spaces. We also study relations among these concepts. Secondly, we introduce the notion of M-fuzzifying base-point, by which we further characterize M-fuzzifying geometric (resp. Peano, Pasch and Sand-glass) property. In addition, we introduce the notion of M-fuzzifying gated amalgamations of M-fuzzifying geometric interval spaces, and discuss several M-fuzzifying gated amalgamation theorems. Finally,we introduce notions of M-fuzzifying convex matroids and M-fuzzifying indepen-dent structures whose one-to-one correspondence is accordingly established.In the theory of (L, M)-fuzzy convex structures, we firstly introduce and characterize domain finiteness of (L, M)-fuzzy convex structures. Secondly, we introduce (L, M)-fuzzy convex matroids and (L, M)-fuzzy independent structures.Then we further establish their relations. Finally, we introduce the notion of(L, M)-fuzzy JHC convex structures, and discuss relations between (L, M)-fuzzy JHC convex structures and (L, M)-fuzzy interval spaces.In the theory of category of (L, M)-fuzzy convex structures, we introduce several subcategories of the category of (L, M)-fuzzy convex structures, such as the category of (L,M )-fuzzy restricted hull spaces, the category of (L,M)-fuzzy stratified convex structures, the category of (L, M)-fuzzy domain finite closure spaces,the category of (L, M)-fuzzy convex structures generated by (L, M)-fuzzy closure structares, the category of (L, M)-fuzzy convex structures generated by M-fuzzifying convex structures, the category of (L, M)-fuzzy weakly induced convex structuring and the category of (L, M)-fuzzy induced convex structures. We discuss their relations and establish several Galois’s correspondences among them.