Research On Several Problems In M-fuzzifying Matroids And M-fuzzifying Convexity Spaces

This thesis mainly includes two parts.Part I includes Chapter 2 and Chapter 3.In Chapter 2,the dual and minors of M-fuzzifying matroids are defined and some properties are discussed.In Chapter 3,the free product of M-fuzzifying matroids is studied.Part II includes Chapter 4 and Chapter 5.In Chapter 4,some characterizations of M-fuzzifying restricted hull operators are obtained and the equivalent notion M-fuzzifying betweenness is also given.As applications,some problems about the cut convex structures and their hull operators of an M-fuzzifying convexity space,the mappings between M-fuzzifying convexity spaces,and the construction of M-fuzzifying convexity spaces are discussed.In Chapter 5,the M-fuzzifying convexity spaces induced by M-fuzzy quasi-orders,M-fuzzy partial orders,and strong M-fuzzy partial orders,respectively are studied.In the following,let me explain explicitly what I have done.Chapter 1 is the foundations of the whole thesis.A concise survey of M-fuzzifying matroids and M-fuzzifying convexity spaces is presented.The necessary notions and results about distributive lattices,fuzzy sets,matroids,convexity spaces,M-fuzzifying matroids and M-fuzzifying convexity spaces are listed.In Chapter 2,the dual and minors of M-fuzzifying matroids are studied.Firstly,a new kind of subtraction of M-fuzzy natural numbers is defined and some basic properties are derived.Secondly,using the new kind of subtraction,the notions of the dual and minors of M-fuzzifying matroids are introduced in the case that M =[0,1]or M is a finite Boolean algebra.Finally,some properties such as the condition of idempotent of the dual of a fuzzifying matroid and the base-map of the dual of an M-fuzzifying matroid are discussed.In Chapter 3,the free product of M-fuzzifying matroids is studied.Firstly,the free product of two M-fuzzifying matroids is defined when M is a finite Boolean algebra.Sec-ondly,a number of properties of the operation are obtained.Finally,the notions of M-fuzzy truncation and M-fuzzy lift are introduced and are used to characterize the minors of the free product of two M-fuzzifying matroids.In Chapter 4,M-fuzzifying restricted hull operators are studied.Firstly,the notion of M-fuzzifying restricted hull operators is given.Then the one-to-one correspondence between M-fuzzifying restricted hull operators and M-fuzzifying convexity spaces is shown.Secondly,the relations between the cut convexity spaces of an M-fuzzifying convexity space and the convexity spaces determined by the cut operators of its M-fuzzifying restricted hull operator are discussed.Finally,the notion of M-fuzzifying betweennesses is given,which is equiv-alent to the notion of M-fuzzifying restricted hull operators.These notions can be used to characterize M-fuzzifying preserving functions and to construct M-fuzzifying convexity spaces.In Chapter 5,M-fuzzifying convexity spaces induced by M-fuzzy quasi-orders,M-fuzzy partial orders,and strong M-fuzzy partial orders are studied.Firstly,using M-fuzzifying restricted hull operator,the notion of M-fuzzifying convexity spaces induced by M-fuzzy quasi-orders can be defined conveniently.Secondly,the notions of M-fuzzifying geometric interval spaces and M-fuzzifying convex geometries are given and then it is shown that M-fuzzifying convexity spaces induced by strong M-fuzzy partial orders are M-fuzzifying geometric interval spaces and M-fuzzifying convex geometries.Finally,the notion of M-fuzzy base-point quasi-order at a point of an M-fuzzifying interval space is introduced,which is used to characterize M-fuzzifying geometric interval spaces.In Chapter 6,conclusions and expectations are made.