【作者】 李小南；

【导师】 李生刚；

【作者基本信息】 陕西师范大学 ， 基础数学， 2006， 硕士

【摘要】 拟阵是组合数学和离散数学的重要组成部分。拟阵的概念已被不同的人用不同的方式加以推广,推广之后的概念主要有偏序集拟阵,广义拟阵和模糊拟阵。由于有限偏序集类和有限分配格类之间存在某种对应关系,我们可以用格的语言建立偏序集拟阵的概念,称之为组合概型。本文中拟阵的推广理论指偏序集拟阵(组合概型)和模糊拟阵。在拟阵理论中,我们可以通过限制,收缩,截短,延伸等运算从一个拟阵生成新的拟阵。本文将主要研究偏序集拟阵(组合概型)和模糊拟阵中的这些生成运算,要点及主要内容如下: 一、给出了限制和收缩作用相等的一个充要条件,显示了秩函数在研究偏序集拟阵中的重要作用。详细的讨论了组合概型中的五种生成运算,并研究了它们的一些性质。在一类特殊的偏序集拟阵中,定义了三种算子,研究了这三种算子的一些性质,证明了类似于拓扑空间中Kuratowski十四集定理的结论。 二、研究了模糊独立集系统的生成运算。举例说明了模糊独立集系统的正规性、基本列与它的限制和收缩的正规性、基本列之间没有必然的联系。定义了模糊拟阵的水平限制、水平截短和垂直截短,证明了模糊拟阵的限制的基本列包含于它的基本列,而这一点对模糊独立集系统来说并不成立。证明了模糊拟阵中圈的传递性定理,为进一步研究模糊拟阵的连通性提供了可能。 三、给出了闭模糊拟阵的一个等价刻画,从而指出了HFM(即Yuang-Cheh Hsuesh在文献[17]中给出的模糊化拟阵)和模糊拟阵的联系。提出了模糊拟阵生成空间的概念,证明了一个模糊拟阵是闭的当且仅当它的生成空间是列紧的。更多还原

【Abstract】 Matroid theory is an important part of combinatorial mathematics and discrete mathematics. It has been generalized and some new theories based, such as poset matroids, greedoids and fuzzy matroids. Because of the one to one correspondence between finite posets and finite distributive lattices, we can give an euivalent defination of poset matroids that uses the language of distributive lattices and call it combinatorial scheme. In this article, the extended theories of matroids are poset matroids (combinatorial scheme) and fuzzy matroids. We can construct new matroids by basic matroid opetations, such as truncation, expansion, restrication and contraction, etc. We will study these operations in poset matroids (combinatorial scheme) and fuzzy matroids. The main content of this paper is as follows:1. This paper gives a necessary and sufficient condition ensuring that the restriction and contraction to same subset of a poset matroid processes the same result, which indicates that the rank function plays an important role in the study of poset matroids. And then we discuss some operations on combinatorial schemes, such as restriction, contraction, truncation and elongation. Some properties of these operations are also studied. This paper gives definitions of the closure operator, interior operator and complement operator in poset matroids, of which the underlying posets have an order-reversing involution, and then studies some properties of these operators. At last, fourteen filters theorem in poset matroids is proved.2. Construtive operations of fuzzy independence set systems are studied. We take an example to say that some aspects are no relations between fuzzy independence set systems and its restriction,such as regularity and fundamental sequences. Vertical truncation, horizontal truncation and vertical restriction of fuzzy matroids are defined. we prove that the fundamental sequences of fuzzy matroids include the fundamental sequences of its restriction, and this is not true to fuzzy independence set systems. Fuzzy circuit transmission theorem is proved, on which we may study the connection of fuzzy matroids.3. A equivalent condition of closed fuzzy matroids is represented, it indicates the connection of HFM and fuzzy matroids. And then, induced space of fuzzy matroids is defined. We proved that a fuzzy matroid is closed if and only if its induced space is sequentially compact space.更多还原