近似有限拟阵
Nearly Finitary Matroids
摘要：
本文通过引入近有限拟阵的新定义来研究近有限拟阵，并证明了其各种性质。2010年，Bruhn等人提出了一个关于无穷拟阵的公理系统，本文使用这个公理系统。在第二章中，我们回顾了历史背景和动机，总结了我们的主要结果。在第三章中，我们定义了拟阵谱的概念。此外，我们证明了一个近有限拟阵的谱可以大于任何固定的有限大小。我们还给出了一个例子，拟阵的无限大的频谱是不是近有限的。假设存在一个拟阵是近有限的但不是$k$ -近有限的，我们构造了一类拟阵是近有限的但不是$k$ -近有限的.我们还证明了有限秩拟阵是可并的。在第四章中，我们将引入近有限化的概念。我们还给出了一个例子，几乎有限的独立系统，是不是$k$ -几乎有限的。这个独立系统不是拟阵。在第5章中，我们将讨论Psi-拟阵并介绍一个可能的推广。此外，我们研究这些新的拟阵，寻找一个例子，几乎有限拟阵是不是$k$ -几乎有限的。我们还没有找到这样的例子。在第六章中，我们将讨论薄和拟阵，并考虑我们的问题仅限于这类拟阵。我们的结果的动机是开放的问题，是否每个几乎有限拟阵是$k$ -几乎有限的一些$k$。
英文摘要：
In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system for this thesis. In Chapter 2, we summarize our main results after reviewing historical background and motivation. In Chapter 3, we define a notion of spectrum for matroids. Moreover, we show that the spectrum of a nearly finitary matroid can be larger than any fixed finite size. We also give an example of a matroid with infinitely large spectrum that is not nearly finitary. Assuming the existence of a single matroid that is nearly finitary but not $k$-nearly finitary, we construct classes of matroids that are nearly finitary but not $k$-nearly finitary. We also show that finite rank matroids are unionable. In Chapter 4, we will introduce a notion of near finitarization. We also give an example of a nearly finitary independence system that is not $k$-nearly finitary. This independence system is not a matroid. In Chapter 5, we will talk about Psi-matroids and introduce a possible generalization. Moreover, we study these new matroids to search for an example of a nearly finitary matroid that is not $k$-nearly finitary. We have not yet found such an example. In Chapter 6, we will discuss thin sums matroids and consider our problem restricted to this class of matroids. Our results are motivated by the open problem concerning whether every nearly finitary matroid is $k$-nearly finitary for some $k$.
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