The structure of 2-separations of infinite matroids

TL;DR

This paper extends a finite matroid decomposition theorem to infinite matroids, establishing a unique tree structure that captures their 2-separations and is invariant under duality.

Contribution

It introduces a unique, duality-invariant tree decomposition for infinite connected matroids based on 2-separations, generalizing finite matroid results.

Findings

Existence of a unique tree decomposition for infinite matroids

Decomposition nodes are minors that are 3-connected, circuits, or cocircuits

Decomposition is invariant under duality

Abstract

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.