The Unravelling Problem

TL;DR

This work studies the unravelling problem for woven families of sets and its lattice and poset analogues, motivated by submodularity in abstract separation systems and tangles. It establishes positive results in two core settings: (i) order-induced, submodular-induced families admit unravellings through a tie-breaking perturbation that preserves submodularity, and (ii) all woven finite posets admit unravellings by iteratively removing elements with a unique cover. It also provides a striking negative result by constructing a non-distributive lattice with a woven subset that cannot be unravelled, and discusses implications for structural submodularity in separation systems. Collectively, the results delineate when unravelling is possible and when it inherently fails, connecting lattice-theoretic and separation-system perspectives.

Abstract

We identify and study a simple combinatorial problem that is derived from submodularity issues encountered in the theory of tangles of graphs and abstract separation systems.

Figures (1)

• Figure 1: The Hasse diagram of $L$. The points in $P$ are denote by black dots, the points outside of $P$ are white.

Theorems & Definitions (43)

• Theorem 2
• Theorem 3
• Proposition 2.2
• proof
• Corollary 2.3
• Corollary 3.2
• proof
• Theorem 4.1
• Lemma 4.2
• proof