Studies On M-fuzzifying Matroids And Their Related Theory

This thesis contains three parts.The first part contains Chapter 2 and Chapter 3.The characterizations of M-fuzzifying matroids and several kinds of M-fuzzifying matroids are given in this part.The second part includes Chapter 4.In this part,the notions of M-fuzzifying matroid(the sense of Vel)and M-fuzzifying independent spaces are defined.The properties are also discussed.The last part includes Chapter 5.The categories of strong M-fuzzifying matroids and strong M-fuzzifying independent spaces are presented in this part.The main research contented in this thesis is as follows.Chapter 1 is introduction.A general survey of this thesis and necessary definitions of lattice theory,fuzzy mathematics,convex structure,matroids are introduced.Chapter 2 is the characterizations of M-fuzzifying matroids.Firstly,by the proper-ties of fuzzifying circuit mapping,the fuzzifying circuit axioms of fuzzifying matroids are presented.Secondly,the base axioms of fuzzifying matroids are obtained.Finally,the prop-erties of M-fuzzifying closure operators are studied.By the M-fuzzifying Exchange Law,the notion of M-fuzzifying matroid closure operators is introduced.It is proved that M-fuzzifying matroid closure operators can be used to induce a kind of M-fuzzifying matroids,which is called strong M-fuzzifying matroids.Conversely,M-fuzzifying matroid closure operators can be induced by strong M-fuzzifying matroids.There is a one-to-one correspondence between M-fuzzifying matroid closure operators and strong M-fuzzifying matroids.Chapter 3 is several kinds of M-fuzzifying matroids.Firstly,M-fuzzifying loop-free matroids and M-fuzzifying simple matroids are defined.The properties of M-fuzzifying rank functions,M-fuzzifying circuit mappings and M-fuzzifying closure operators of this two types are studied.Secondly,fuzzifying paving matroids and fuzzifying uniform matroids are presented.It is shown that fuzzifying uniform matroids,fuzzifying matroids with R(E)? 1,fuzzifying loop-free matroids with R(E)<2 and fuzzifying simple matroids with R(E)?3 are all fuzzifying paving matroids.Finally,the independent set axioms,fuzzifying circuit xioms and fuzzifying base axioms of fuzzifying paving matroids are discussed.Chapter 4 is M-fuzzifying independent spaces.Firstly,the relation between indepen-dent spaces and loop-free matroids are discussed.The notion of simple matroids(the sense of Vel)and simple independent spaces are introduced.Secondly,the properties of M-fuzzifying closure operators and M-fuzzifying convex hull operators are studied.By M-fuzzifying Ex-change Law,the concept of M-fuzzifying matroids(the sense of Vel)are defined.It is proved that an M-fuzzifying convexity preserving mapping and M-fuzzifying convex-to-convex map-ping image of an M-fuzzifying matroid is also an M-fuzzifying matroid.Finally,the notions of M-fuzzifying independent spaces and strong M-fuzzifying independent spaces are introduced.Strong M-fuzzifying independent spaces can be induced byM-fuzzifying matroids(the sense of Vel),and conversely,by the notion of M-fuzzifying flags,it is shown that M-fuzzifying matroids(the sense of Vel)can be induced by strong M-fuzzifying independent spaces.It is also proved that F?C = C and ?F? = ?.Chapter 5 is the categories of strong M-fuzzifying matroids and strong M-fuzzifying independent spaces.Firstly,preliminaries about category theory are given.Secondly,by the concept of M-fuzzifying strong mappings,the category of strong M-fuzzifying matroids,the category of strong M-fuzzifying loop-free matroids and the category of strong M-fuzzifying simple matroids are established,which are denoted by S-MF-M,S-MF-LFM and S-MF-SM,respectively.When M = {T,_L},those categories are denoted by M,LFM and SM,respectively.It is proved that M is a reflective subcategory and coreflective subcategory of S-MF-M.That is M r,c(?)S-MF-M.Similarly,it is shown that LFM r,c(?)S-MF-LFMand SM r,c(?)S-MF-SM.Thirdly,the category of strong M-fuzzifying independent spacesS-MF-IS is given.It is proved that S-MF-IS is isomorphic to the category of M-fuzzifying matroids(the sense of Vel),which is denoted by MF-VM.When M = {(?),?},IS is denoted the category of independent spaces and VM is denoted the category of matroids(the sense of Vel).It is proved that IS is a reflective subcategory of S-MF-IS and VM is a reflec-tive subcategory of MF-VM.The last of this chapter shown that S-MF-LFM is an full subcategory of S-MF-IS.Chapter 6 is conclusion.