Studies On (L,M)-Fuzzy Convexity Spaces And Their Related Theory

This thesis mainly includes three parts. Part I includes Chapter 2 and Chapter 3. In this part, (L,M)-fuzzy convexity spaces and M-fuzzifying interval spaces are defined and investigated.Part Ⅱ is Chapter 4. In this part, M-fuzzifying submodular functions are studied and the relations between M-fuzzifying submodular functions and M-fuzzifying matroids is discussed. Part Ⅲ is Chapter 5. In this part, categories of (L, M)-fuzzy greedoids are studied. In the following, let me explain explicitly what I have done.Chapter 1 is the foundations for the whole thesis. In this chapter, some preliminaries, which will be used throughout this thesis, that related with lattice theory and fuzzy sets, that with category theory (especially concrete category) and that with matroids as well as (L,M)-fuzzy matroids are recalled and listed.In Chapter 2, (L,M)-fuzzy convexity spaces are studied. Firstly, for completely dis-tributive lattices L and M, axioms of (L, M)-fuzzy convexity spaces are given. Secondly, under the framework of (L, M)-fuzzy convexity spaces, some basic concepts are defined in-cluding (L, M)-fuzzy convexity preserving functions, (L, M)-fuzzy convex-to-convex function-s, bases, subbases, quotient spaces, subspaces, disjoint sums, products and joins. Thirdly, based on a completely distributive lattice L with β(a∧b)=β(a)∩β(b) for a,b ∈ L, the categorical relationship between (L, M)-fuzzy convexity spaces and M-fuzzifying convexity spaces is discussed. It is shown that there exists an adjunction between MYCS and LMCS, where MYCS and LMCS denote the category of M-fuzzifying convexity spaces, and the category of (L, M)-fuzzy convexity spaces, respectively. At last, for a completely distributive lattice M with an order-reversing involution, axioms of M-fuzzifying hull operators are given and it is proved there is a one-to-one correspondence between M-fuzzifying hull operators and M-fuzzifying convexity spaces.In Chapter 3,M-fuzzifying interval spaces are defined and studied and then the categorical relationship between M-fuzzifying interval spaces and M-fuzzifying convexity s- paces is discussed. Firstly, M-fuzzifying interval spaces, M-fuzzifying interval operators and M-fuzzifying interval preserving functions are introduced and the relationship between M-fuzzifying interval spaces and M-fuzzifying convexity spaces is discussed. It is shown that an M-fuzzifying convexity space is induced by an M-fuzzifying interval space if and only if it is of M-fuzzifying arity≤2. Secondly, (L,M)-fuzzy convexity preserving functions and (L,M)-fuzzy convex-to-convex functions are characterized by M-fuzzifying hull operators and it is proved that MYCSA2 is a coreflective full subcategory of MYQS, where MYCSA2 and MYQS denote the category of M-fuzzifying convexity spaces of M-fuzzifying arity≤2 and the category of M-fuzzifying interval spaces, respectively. Finally, subspaces and product spaces of M-fuzzifying interval spaces are presented and their fundamental properties are obtained.In Chapter 4, the concept of M-fuzzifying submodular functions are defined and stud-ied and the relations between and M-fuzzifying submodular functions M-fuzzifying matroids are discussed. Firstly, the concept of M-fuzzifying submodular functions is introduced. It is shown that an M-fuzzifying matroid can be obtained from an M-fuzzifying submodular function in three ways, that is, a circuit-map, an M-fuzzifying family of independent sets and an M-fuzzifying family of dependent sets can be induced by an M-fuzzifying submodular function. Secondly, circuit-maps and base-maps are induced by M-fuzzifying family of inde-pendent sets and then characterizations of base-maps and circuit-maps are obtained by means of M-fuzzifying submodular function. Thirdly, the notion of the union of M-fuzzifying matri-ods is introduced and some of its properties related to M-fuzzifying restrictions, M-fuzzifying direct sums and M-fuzzifying submodular functions are discussed. At last, generalizations of Edmonds’Intersection Theorem and Edmonds’Covering Theorem are established in the framework of M-fuzzifying matriods.In Chapter 5, the categorical relations among greedoids, [0,1]-greedoids, fuzzifying greedoids and (I,I)-fuzzy greedoids are studied. Greedoids and feasibility preserving map-pings form a category, which is denoted by G. [0,1]-greedoids and [0,1]-feasibility preserving mappings form a category, which is denoted by FG. Closed and perfect [0,1]-greedoids and [0,1]-feasibility preserving mappings form a category, which is denoted by CPFG. Fuzzifying greedoids and fuzzifying feasibility preserving mappings form a category, which is denoted by FYG. (I,I)-fuzzy greedoids and (I,I)-fuzzy feasibility preserving mappings form a category, which is denoted by BIFG. The main results are summarized as follows:G(?)cFG,FYG(?)cBIFG,G(?)r,cFYG(?)CPFG(?)FG(?)r,cBIFG, where r, c mean reflective and coreflective, respectively.In Chapter 6,conclusion remarks and expectation are made.