【作者】 毛华；

【导师】 刘三阳；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2002， 博士

【摘要】 随着科学技术的进步特别是计算机的诞生和发展，起源于20世纪30年代的拟阵理论，其原有的结构已不能满足新的要求，拓广原有的拟阵结构势在必行。拓广的途径多种多样，最著名和最有代表性的：一是将拟阵定义的底集用偏序集替代而产生的偏序集拟阵，二是满足拟阵的公理的有序形式的集合—广义拟阵。时至今日，二者仍在不断发展。 从拟阵的发展历史可以看出，正是由于从不同的角度出发去研究拟阵，才使得拟阵在更广的范围内得到应用。另外，在偏序集拟阵论和广义拟阵论的产生、发展的道路上，也显示出偏序集理论所扮演的重要角色。因此，本文采用与前人不同的思维方式，利用偏序集理论，在保持拟阵某些原有性质的基础上，对两种新理论给予充实和完善。同时讨论两种新理论的区别与联系。 由于反拟阵是与拟阵“对立”的一种广义拟阵，所以在考虑广义拟阵的性质时，必不可少地成为考虑和研究对象。 此外，这里还将拟阵的方法运用到在数据挖掘和数据分析中非常有效的数学工具之一—概念格上，为概念格的研究提供了新思路和新方法。 本文主要结果如下： (1)拟阵中有关截短、延长、约束和收缩这四种子运算分别在保留其原有性质的基础上被推广到偏序集拟阵理论，利用它们讨论了偏序集拟阵的连通性；它们与其它算子如闭包算子等相结合，得出偏序集拟阵为广义拟阵而反之不然的结论。 (2)在将拟阵自同构的概念推广至偏序集拟阵、反拟阵和广义拟阵之后，利用群论的方法证得：偏序集拟阵不是反拟阵的推广，而拟阵与反拟阵也无互相包含关系。结合(1)，进一步说明广义拟阵包含偏序集拟阵。 (3)从偏序集理论角度出发，进一步探讨了偏序集拟阵与广义拟阵的关系，证明偏序集拟阵为区间广义拟阵而反之不然。 用偏序集理论给出了讨论不同类广义拟阵之间关系的方法。 (4)虽然拟阵间强映射的概念可推广到偏序集拟阵、反拟阵和广义拟阵，但是拟阵间强映射具有的交换定理在三者对应的性质上却反映不同，它们均不具有交换定理。 然而，对给定的一个反拟阵及广义拟阵，其强映射的寻求却可在偏序集理论支持下找到多种方法。 研究表明，尽管反拟阵间的强映射为弱映射，而反之不然，但反拟阵间的弱映射是不能取代强映射的.这可以从二者对交换定理的反映上看出.此结论也表明研究强映射的必要性. 由于区间广义拟阵是广义拟阵中应用广泛的一个大家族，故文中专门给出了关于它的自同构的判定方法. (5)运用先分类、后分层的办法，对给定有限集上的全体拟阵提出了一种构造方法;借助偏序集理论，建立起给定有限集上全体拟阵与图论中树的联系，从而得到了另一种寻找给定有限集上全体拟阵的构造方法.这两种方法使得拟阵可以运用到寻找概念格的研究中，也使得拟阵理论与概念格理论的关系更为密切.更多还原

【Abstract】 Matroids were invented around 1930. Unfortunately, the classical notions of matroids are not suited to the contemporary commands with the advent and development of scientific technology especially computer science. Based on this, the work of the generalization of the classical matroids is imperative. This work is done in many ways, the most famous and representative are the following i one is poset matroid that comes from replacing the underlying set of a matroid by a partially ordered set; another is greedoid defined as a collection satisfying an ordered version of the matroid axioms. Even today, the new two theories are developing continuously.A look at the history of matroids might lead to the conclusion that just studied from different directions, matroid theory has been rich in connections with pure and applied, deeply rooted in the utmost reaches of experimental thinking. In addition, for the theory of both poset matroids and greedoids, poset theory was one of the key sources. So it is not surprising that in this dissertation, with the help of poset theory, the two new theories are polished and refined in different methods that are distinct from former scholars. At the same time, some properties of matroids are extended in many ways without losing some of the style and features of these concepts. Some relationships between the two new theories are discussed too. All these are polishing and refining on the new two theories.Antimatroids as a class of greedoids are in many respects "antipodal" to matroids. Thus it is natural to consider and study on antimatroids when research into the properties of greedoids.Besides, it is known that concept lattice is one of an effective mathematical tool for the study of data mining and data analysis. The application of matroid theory into it is a new idea and a new method in the research of concept lattice.The main results as follows.(1) Four submatroids -truncation, elongation, restriction and contraction are extended from matroids to poset matroids respectively without losing some of the valuable properties of them. In addition, combining with other operators such as closure operators, the new four subposetmatroids are applied into the discussion about the connectivity in poset matroids and give a conclusion that every poset matroid is agreedoid, but not vice versa.(2) The concepts of the automorphisms of a poset matroid, an antimatroid and a greedoid are common generalizations of that of a matroid. By dint of group theory, some results are obtained: one is that poset matroids are not the extension of antimatroids; another is that the inclusion relations between matroids and antimatroids are not existed. Combining with one of the results in (1), it follows that the field of greedoids owned is larger than the part of poset matroids.(3) Poset theory has given rise to a comprehensive view of the relation between poset matroids and greedoids deeply. The comprehension is that poset matroids are interval greedoids, but not vice versa.Here, for different classes of greedoids, a method for dealing with the relations among diem comes into life by poset theory.(4) One of the most beautiful results of matroids is Exchange Theorem relating the strong maps between matroids: However, the correspondent result is not true for poset matroids, antimatroids and greedoids respectively.Even then some methods for searching the strong maps of a given antimatroid and a given greedoid are found out using poset theory.Though strong maps "of an antimatroid are weak and not the converse, fortunately, strong is not to be replaced by weak in virtue of their different attitudes to the Exchange Theorem. This result shows the importance to notice the strong maps.Some decision methods are given for the automorphisms of interval greedoids that are a very large class of greedoids.(5) A constructive way is given for obtaining the whole matroids defined on the same finite ground by classifying first and stratifying second. After building the relation between a tre更多还原