Relative rank axioms for infinite matroids

TL;DR

This paper introduces a new system of axioms based on relative rank for infinite matroids, highlighting limitations of rank functions and expanding the theoretical framework of matroid theory.

Contribution

It proposes a novel set of axioms using relative rank to characterize infinite matroids, addressing gaps in existing axiomatizations.

Findings

Rank functions may not fully characterize infinite matroids.

A new axiomatic system based on relative rank is proposed.

The system simplifies understanding of infinite matroids.

Abstract

In a recent paper, Bruhn, Diestel, Kriesell and Wollan (arXiv:1003.3919) present four systems of axioms for infinite matroids, in terms of independent sets, bases, closure and circuits. No system of rank axioms is given. We give an easy example showing that rank function of an infinite matroid may not suffice to characterize it. We present a system of axioms in terms of relative rank.