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

TL;DR

The paper addresses canonical tree-of-tangles results for structurally submodular separation systems by introducing $\mathcal{P}$-submodularity and proving the existence of a canonical nested tree set $N(\vS,\mathcal{P})$ that distinguishes a given family of orientations $\mathcal{P}$. The method avoids reliance on ambient lattice embeddings and ensures invariance under isomorphisms, strengthening previous results by delivering canonicity. The main technique constructs canonical representatives from $M_P$, uses their infima to produce a nested set, and then applies induction on the remaining orientations to assemble the final canonical tree set. This has implications for reproducible algorithms and broad applicability of tree-of-tangles results beyond graph settings. The work thus extends the reach of tangle theory to more general separation systems while preserving a canonical, input-invariant construction.

Abstract

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

Theorems & Definitions (17)

• Theorem 1: AbstractTangles*Theorem 6
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5: See also AbstractSepSys*Lemma 3.2
• proof
• Lemma 6
• proof : Proof of \ref{['lem:nonempty']}.
• Lemma 7