Order (2018) 35:157–170
Abstract Separation Systems
Received: 21 December 2015 / Accepted: 27 February 2017 / Published online: 25 April 2017
© The Author(s) 2017.
Abstract Abstract separation systems provide a simple general framework in which both
tree-shape and high cohesion of many combinatorial structures can be expressed, and their
duality proved. Applications range from tangle-type duality and tree structure theorems in
graphs, matroids or CW-complexes to, potentially, image segmentation and cluster analysis.
This paper is intended as a concise common reference for the basic definitions and facts
about abstract separation systems in these and any future papers using this framework.
Keywords Connectivity · Graph · Tangle · Lattice · Partial order · Tree
Formally, abstract separation systems are very simple objects: posets with an order-reversing
involution. Think of the oriented separations (A, B) of a graph, where (A, B) ≤ (C, D) if
A ⊆ C and B ⊇ D, and the involution is given by (A, B) → (B, A).
What makes such ‘separation systems’ of graphs immediately interesting is that what
little information they capture from the structure of a graph suffices to express, and to prove,
two of the central theorems in graph minor theory: the tangle-tree theorem and the tangle
duality theorem of Robertson and Seymour . Even just for graphs this is not obvious.
But in fact, it can be done much more generally.
Using abstract separation systems, one can prove tangle-tree  and tangle duality 
theorems that apply to other combinatorial structures too. In each of these, ‘tangles’ can
encode bespoke cohesive substructures of that structure. The tangle-tree theorem shows how
these can be separated in a tree-like way (thus, decomposing the given structure into its
Hamburg University, Hamburg, Germany