Axioms for infinite matroids

Abstract

We give axiomatic foundations for infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. Continuing work of Higgs and Oxley, this completes the solution to a problem of Rado of 1966. (C) 2013 Henning Bruhn, Reinhard Diestel, Matthias Kriesell, Rudi Pendavingh and Paul Wollan. Published by Elsevier Inc. All rights reserved

Orthogonality and Dimensionality

“…First, even though the study of infinite matroids has long been problematic due to the lack of a unified approach capturing all the important aspects of finite matroid theory, it appears that recently, a satisfactory approach has been obtained [22], where an additional axiom (M ) is introduced, stating that for every subset X of a matroid E and every independent subset I ⊆ X, the set of independent subsets J such that I ⊆ J ⊆ X has a maximal element. The obtained structure is that of B-matroids [23] which are known to have equicardinal bases [24].…”

Factorization theorems for strong maps between matroids of arbitrary cardinality

“…[1][2][3][4][5], [6,Ch.7], [17,Ch.8]). The most well known results for strong maps in the category of finite matroids are factorization theorems.…”

“…In this paper we used the definition of infinite matroid given in [9]. Comparing the definition of infinite matroid given in [9] with that in [17] (i.e. finitary matroids in [17]), we find that the definition in [17] is more general than that in [9].…”

“…Comparing the definition of infinite matroid given in [9] with that in [17] (i.e. finitary matroids in [17]), we find that the definition in [17] is more general than that in [9]. In addition, the authors of [17] not only compared the properties of finite matroids with those of 'B-matroids' by Oxley [8], but also sought to find a formulation of the axioms of infinite matroids.…”

“…In addition, the authors of [17] not only compared the properties of finite matroids with those of 'B-matroids' by Oxley [8], but also sought to find a formulation of the axioms of infinite matroids. Hence, in future work we may consider the factorization theorems for strong maps between infinite matroids which satisfy the axioms of a finite matroid given in [17].…”