Infinite Matroids

解决了Rado(1966)的一个长期存在的问题，第一个被点名的申请人和合著者最近证明了无限拟阵可以公理化为类似于有限拟阵。在20世纪80年代之前的一系列此类尝试失败后，人们普遍认为这可能是不可能的。特别是，人们认为具有通常性质的基和电路与拟阵对偶性是不相容的。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，该公理确保了基和圈的存在，但排除了对偶：一个无限集只要它的所有有限子集都是独立的，它就是独立的。由于这个公理而导致的对偶的缺乏，实际上阻碍了任何无限拟阵理论的发展，就像他们的有限理论一样。新发现的无限拟阵的公理(具有对偶)将首次使研究大类已知的非有限拟阵成为可能，例如有限拟阵的对偶。此外，我们希望找到以前被有限公理排除的一般无限的应用，例如Banach或Hilbert空间。

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.