Structural submodularity and tangles in abstract separation systems

Abstract

We prove a tree-of-tangles theorem and a tangle-tree duality theorem for abstract separation systems → S that are submodular in the structural sense that, for every pair of oriented separations, → S contains either their meet or their join defined in some universe U of separations containing → S . This holds, and is widely used, if U comes with a submodular order function and → S consists of all its separations up to some fixed order. Our result is that for the proofs of these two theorems, which are central to abstract tangle theory, it suffices to assume the above structural consequence for → S , and no order function is needed.Abstract separation systems were first introduced in [8]; see there for a gentle formal introduction and any terminology we forgot to define below. Motivation for why they are interesting can be found in the introductory sections of [11,13,12] and in [14]. In what follows we provide a self-contained account of just the definitions and basic facts about abstract separation systems that we need in this paper.A separation system ( → S , ≤, * ) is a partially ordered set with an order-reversing involution * :S is → s * , which we usually denote by ← s . An (unoriented) sepas separation systems. For separation universes, therefore, submodularity is used with the narrower meaning of being endowed with a submodular order function [8].

A canonical tree-of-tangles theorem for structurally submodular separation systems

“…For a full introduction to abstract tangle theory and its terminology and notation we refer the reader to [2,4]. In the remainder of this section we offer a brief introduction of only those terms and notation of tangle theory that are relevant to this paper.…”

“…Later Diestel, Erde, and Weißauer [4] showed that the latter structural condition by itself is already sufficiently strong for proving tree-of-tangles theorems: tangle theory can be meaningfully studied without the hitherto usual assumption of a submodular order function, further widening its applicability. If a separation system has this structural property but not necessarily a submodular order function then it is structurally submodular or simply submodular if the context is clear.…”

“…If a separation system has this structural property but not necessarily a submodular order function then it is structurally submodular or simply submodular if the context is clear. The tree-of-tangles theorem established in [4] then reads as follows: Theorem 6]). Let S be a structurally submodular separation system and P a set of profiles of S. Then S contains a tree set N that distinguishes P.…”

“…This Theorem 1.1 is even more widely applicable than the result of [5], but has one major downside: it does not yield canonicity, since the proof in [4] chooses certain separations arbitrarily.…”

Duality Theorems for Blocks and Tangles in Graphs

“…Theorem 15. Let − → S be a separable 4 submodular separation system in some universe of separations, let F ⊆ 2 − → S contain P S , and let F * be any uncrossing of F. Then the following are equivalent: 4 Whilst the assumption that − → S is separable is necessary to apply Theorem 5, in a forthcoming paper [6] the authors and Weißauer show that every submodular separation system is in fact separable, and so this asssumption can be removed from Theorem 15.…”