Structural submodularity and tangles in abstract separation systems

TL;DR

This paper establishes tangle-tree and tangle duality theorems for abstract separation systems that are structurally submodular, removing the need for an order function in the proofs.

Contribution

It proves that the core theorems of abstract tangle theory hold under a structural submodularity condition without requiring an order function.

Findings

Proves tangle-tree theorem for structurally submodular separation systems.

Establishes tangle duality theorem without order functions.

Shows core tangle theory results depend only on structural submodularity.

Abstract

We prove a tangle-tree theorem and a tangle duality theorem for abstract separation systems $\vec S$ that are submodular in the structural sense that, for every pair of oriented separations, $\vec S$ contains either their meet or their join defined in some universe $\vec U$ of separations containing $\vec S$. This holds, and is widely used, if $\vec U$ comes with a submodular order function and $\vec S$ 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 $\vec S$, and no order function is needed.