Pseudoflowers in infinite connectivity systems

TL;DR

This paper extends the theory of flowers in connectivity systems to infinite cases, establishing the existence of maximal flowers and characterizing their types, thereby deepening the understanding of infinite connectivity structures.

Contribution

It generalizes the concept of flowers to infinite connectivity systems, proves the existence of maximal flowers, and characterizes infinite daisies and anemones in infinite matroids.

Findings

Maximal generalised flowers exist in infinite connectivity systems.

In infinite matroids, only anemones can be extended to infinite objects.

Provides a characterization of when infinite daisies exist.

Abstract

Given a graph or a matroid, a tree of tangles is a tree decomposition that displays the structure of the connectivity: every edge of the decomposition tree induces a separation, that is, a way to divide the graph or matroid into two parts; and for every two highly connected areas (encoded as tangles) that live on different sides of some separation, some separation induced by an edge distinguishes them. Separations induced by a tree of tangles cannot cross. One approach to display even more connectivity structure is to insert even more structure into a tree of tangles, for example the flowers that were introduced by Oxley, Semple and Whittle in 2007 for matroids and generalised to finite connectivity systems by Clark and Whittle in 2013. Most of the separations displayed by a flower are crossing. In order to extend this theory to the infinite case, we generalise the notion of flowers to infinite connectivity systems, and show that there are maximal generalised flowers. Also, we show in the special case of infinite matroids that of the two types of flowers (anemones and daisies) only anemones can be extended to truly infinite objects, and provide for general connectivity systems a characterisation of when infinite daisies exist. Furthermore we describe a more abstract view on the interaction of tangles and separations distinguishing them, which among other things provides additional motivation for why there should be maximal generalised flowers.