On Matroids which have Precisely One Basis in Common

On Ideals Which Have the Weakly Insertion of Factors Property

A one-sided ideal of a ring has the insertion of factors property (or simply, IFP) if implies r for . We say a one-sided ideal of has the weakly IFP if for each , implies , for some non-negative integer . We give some examples of ideals which have the weakly IFP but have not the IFP. Connections between ideals of which have the IFP and related ideals of some ring extensions a...

On Disjoint Common Bases in Two Matroids

We prove two results on packing common bases of two matroids. First, we show that the computational problem of common base packing reduces to the special case where one of the matroids is a partition matroid. Second, we give a counterexample to a conjecture of Chow, which proposed a sufficient condition for the existence of a common base packing. Chow’s conjecture is a generalization of Rota’s ...

On basis-exchange properties for matroids

We give a counterexample to a conjecture by Wild about binary matroids. We connect two equivalent lines of research in matroid theory: a simple type of basis-exchange property and restrictions on the cardinalities of intersections of circuits and cocircuits. Finally, we characterize direct sums of series-parallel networks by a simple basis-exchange property. In [9], Wild proposed a characteriza...

Basis-Transitive Matroids

We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basisand point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the s...

On the matroids in which all hyperplanes are binary