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

Abstract

We show that every structurally submodular separation system admits a canonical tree set which distinguishes its tangles.

Refining trees of tangles in abstract separation systems I: Inessential parts

“…Note that the nested set from [12] satisfies the assumptions of Theorem 3.6, as we will show in Section 4.…”

“…But if we want to apply Lemma 3.5 to some canonical tree set in an arbitrary submodular separation system to obtain a similar result as in Erde's Refinement Theorem, we first have to find such a set that satisfies the assumptions of Lemma 3.5. For this, we will show in Section 4 that the canonical tree set which Elbracht and Kneip [12] constructed satisfies those assumptions. This will then allow us to prove Theorem 1, by refining this set.…”

“…This will allow us to refine its inessential nodes which then implies Theorem 1. For this, throughout this section, let S be a submodular separation system, and let P be a set of profiles of S. We first recall the construction of N (S, P) from [12]: Construction 4.1. A separation s ∈ S is exclusive (for P) if it is contained in exactly one profile in P. If P ∈ P is that profile, we say that s is P -exclusive (for P).…”

“…Set s P := inf(M P,1 ) := r∈MP,1 r for all P ∈ P 1 , set N 1 := {s P : P ∈ P 1 }, and let S 2 be the set of all separations in S which are nested with N 1 . For consistency set also S 1 := S. Then S 2 is still submodular and distinguishes all remaining profiles in P P 1 [12,Lemma 4.5]. Hence, we can proceed in the same manner as follows.…”

“…We then obtain the desired nested set by setting N (S, P) := N K where K is the smallest number for which P >K contains at most one profile. By construction, N (S, P) distinguishes all profiles in P, and it is shown in [12] that N (S, P) is indeed canonical. This completes the construction.…”