2019
DOI: 10.1007/s11083-019-09499-y
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Profinite Separation Systems

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. IntroductionThis paper is… Show more

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Cited by 6 publications
(28 citation statements)
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“…1 Even though − → S itself is not usually profinite in the sense of [4], the topology we define on − → S is the subspace topology of − → S as a subspace of the (profinite) system of all oriented separations of G, equipped with the inverse limit topology from [4].…”
Section: S | Every Ray Of ω Has a Tail In B}mentioning
confidence: 99%
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“…1 Even though − → S itself is not usually profinite in the sense of [4], the topology we define on − → S is the subspace topology of − → S as a subspace of the (profinite) system of all oriented separations of G, equipped with the inverse limit topology from [4].…”
Section: S | Every Ray Of ω Has a Tail In B}mentioning
confidence: 99%
“…Our research expands on this latter result. Every end ω of a graph induces not only a tangle of infinite order in G, but for each k ∈ N the end ω induces a k-tangle in G. The set S k are closed for some, or even all, k ∈ N. In this paper we characterize the ends of G by the behaviour of their tangles, as follows: which these properties tend to generalize can be identified, however: they are the sets of separations that are closed in a certain natural topology [4]. Let us define this topology next.…”
Section: Introductionmentioning
confidence: 99%
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“…This paper is a sequel to, and assumes familiarity with, two earlier papers [2,5]. The first of these [2] introduced finite abstract separation systems, whereas the latter [5] laid the foundations for extending the theory of separation systems to a broad class of infinite separation systems.…”
Section: Introductionmentioning
confidence: 99%
“…In [5] the foundations were laid for extending the tree-of-tangles theorem and the tangle-tree duality theorem to infinite separation systems: [5] introduced, and studied, separation systems that are profinite -i.e. which are determined by the finite separation systems they induce.…”
Section: Introductionmentioning
confidence: 99%