Abstract

Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory about infinite separation systems and how they relate to the finite separation systems they induce. They can be used to prove tangle-type duality theorems for infinite graphs and matroids, which will be done in future work that will build on this paper.

Highlights

  • This paper is a sequel to, and assumes familiarity with, an earlier paper [2] in which finite abstract separation systems were introduced

  • The profinite separation systems introduced here, and the results proved, will form the basis for our proof of the tangle duality theorem for infinite graphs [1], as well as for a more comprehensive study of profinite tree sets [8]

  • Abstract separation systems were introduced in [2] to lay the foundations for a comprehensive study of how tangles, originally introduced by Robertson and Seymour [10] in the course of their graph minors project, can be generalized to capture, and relate, various other types of highly cohesive regions in graphs, matroids and other combinatorial structures

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Summary

Introduction

This paper is a sequel to, and assumes familiarity with, an earlier paper [2] in which finite abstract separation systems were introduced. The basic idea behind tangles is that they describe such a region indirectly: not by specifying which objects, such as vertices or edges, belong to it, but by setting up a system of pointers on the entire structure that point towards this region The advantage of this indirect approach is that such pointers can locate such a highly cohesive region even when it is a little fuzzy – e.g., when for every low-order separation of a graph or matroid ‘most’ of the region will lie on one side or the other, so that this separation can be oriented towards it and become a pointer, but each individual vertex or edge (say) can lie on the ‘wrong’ side of some such separation.

Separation Systems
Inverse Limits of Sets
Let P be the set of all finite subsets of V
Profinite Nested Separation Systems
The Tree Set Compactness Theorem
Orienting Profinite Nested Separation Systems
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