【作者】 路玲霞；

【导师】 李生刚；

【作者基本信息】 陕西师范大学 ， 基础数学， 2005， 硕士

【摘要】 与其他数学分支相比,拟阵理论并不是一个具有悠久历史的古老分支。1935年,Whitney在分析了向量的线性相关性的抽象性质后,建立了拟阵的概念。1988年,R.Goetschel和W.Voxman将“模糊”概念引入拟阵理论,开创了“模糊拟阵”的研究。在此基础上,本文基于最简单的非平凡Fuzzy格L={0,1/2,1},给出了L-模糊拟阵的概念,并研究了L-模糊拟阵的一系列性质。 下面介绍本文的结构和主要内容。 第一章 对文章中将要用到的有关模糊数学与拟阵的基本知识和基本结论作了一个简要的叙述。 第二章 给出了L-模糊拟阵的概念,并由L-模糊拟阵出发,构造了一些新的拟阵和新的L-模糊拟阵,如截拟阵、L-模糊拟阵的直和、L-模糊拟阵的限制、均匀L-模糊拟阵、初等L-模糊拟阵等等。随后研究了L-模糊拟阵和它的截拟阵之间的关系,以及一个模糊集成为独立集的充要条件,得到了一些不错的结论,为后面几章的研究铺平了道路。 第三章 主要研究了L-模糊拟阵的模糊基与模糊圈。首先,给出了模糊基的定义,研究了一个独立集成为模糊基的充要条件,以及模糊基与截拟阵的基的关系;其次,给出了闭L-模糊拟阵与正则L-模糊拟阵的概念,得到了几个主要的结论,如: (1) L-模糊拟阵中的模糊基的支撑集具有相等的势; (2) 正则L-模糊拟阵中,一个模糊集成为模糊基的充要条件是它的截集是对应截拟阵的基; (3) L-模糊拟阵是正则的等价于模糊基具有相等的势。最后,讨论了L-模糊拟阵的模糊圈,研究了模糊集成为模糊圈的充要条件,以及模糊圈与模糊基的关系。 第四章 研究了L-模糊拟阵的模糊秩函数与模糊闭包算子。首先,定义了模糊秩函数,给出了模糊秩函数满足的一些性质;在此基础上,补充了模糊秩函更多还原

【Abstract】 Matroid theory is not an old branch with a long history in comparison with other mathematical branches. In 1935, Whitney analyzed abstract properties of linear dependence of vectors, and established the concepts of matroids. In 1988, R. Goetschel and W. Voxman introduced " fuzzy " concepts into matroid theory, and then started the research of fuzzy matroids. In this thesis, we introduced concepts of L-fuzzy sets and L-fuzzy matroids, and studied main properties and characteristics of L-fuzzy matroids, where L is the simplest nontrivial fuzzy set.The construction and the main contents of this thesis are as follows.In chapter 1, we maked a depiction for basic knowledge and basic conclusions of fuzzy mathematics and matroids which would be used in the foUowing chapters.In chapter 2, the concept of L-fuzzy matroids was given, and then some new matroids and new .L-fuzzy matroids were constructed from the given L-fuzzy matroids, such as cut matroids, direct sum of L-fuzzy matroids, restriction of L-fuzzy matroids, uniform L-fuzzy matroids, elementary L-fuzzy matroids, etc. Then we studied the relations between L-fuzzy matroids and their cut matroids, and the equivalent condition for a fuzzy set to be an independent set. Some good conclusions were obtained, which paved the way for research of the following chapters.In chapter 3, the fuzzy bases and fuzzy circuits of L-fuzzy matroids were mainly studied.Firstly, the definition of fuzzy bases, and the necessary and sufficient conditions for a fuzzy set to be a fuzzy base, and the relations between fuzzy bases and the bases of the cut matroids were given. Secondly, we defined the concepts of closed L-fuzzy matroids and regular L-fuzzy matroids, and then got several main conclusions, as follows:(1) The support sets of bases of an L-fuzzy matroid have the same cardinality.(2) In regular If-fuzzy matroids, the necessary and sufficient condition for a fuzzy set to be a fuzzy base is that the cut set of the fuzzy set is the base of the corresponding cut matroid.(3)A necessary and sufficient condition for a L-fuzzy matroid to be regular is that the bases of L-fuzzy matroid have the same cardinality.更多还原