Infinite Matroids

无限拟阵

• 批准号: 191164225
• 负责人: Professor Dr. Reinhard Diestel
• 金额: --
• 依托单位: Fachbereich Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/191164225?language=en
• 关键词: Infinite; Matroids

Solving a long-standing problem of Rado (1966), the first named applicant and coauthors have recently shown that infinite matroids can be axiomatized similarly to finite matroids. Following a series of unsuccessful such attempts until the 1980s, it had become a commonly held belief that this might be impossible. In particular, it was thought that having bases and circuits with their usual properties was incompatible with matroid duality. As a consequence, most authors either disregarded infinite matroids altogether, or imposed a restrictive additional axiom which ensured that bases and circuits existed, but which ruled out duality: the 'finitary axiom' thatAn infinite set is independent as soon as all its finite subsets are independent.The lack of duality resulting from this axiom in effect prevented the development of any theory of infinite matroids along the lines of their finite theory. The newly found axioms for infinite matroids (with duality) will make it possible for the first time to study large classes of known non-finitary matroids, such as the duals of finitary ones. In addition, we hope to find generically infinite applications previously precluded by tlie finitary axiom, such as to Banach or Hilbert spaces.

解决Rado（1966）的一个长期存在的问题，第一个命名的申请人和合著者最近证明了无限拟阵可以类似于有限拟阵被公理化。直到1980年代，在一系列不成功的尝试之后，人们普遍认为这可能是不可能的。特别是，有人认为，有基地和电路与其通常的性质是不相容的拟阵对偶。因此，大多数作者要么完全忽略了无限拟阵，要么强加了一个限制性的附加公理，以确保基和圈的存在，但排除了对偶性："有限公理“，即一个无限集是独立的，只要它的所有有限子集是独立的。缺乏对偶性导致从这个公理实际上阻止了任何理论的发展，无限拟阵沿着他们的路线，有限理论新发现的无穷拟阵公理（具有对偶性）将首次使研究大类已知的非有限拟阵成为可能，例如有限拟阵的类。此外，我们希望找到一般无限的应用，以前排除了tlie有限公理，如Banach或希尔伯特空间。