Separations of sets
Nathan Bowler, Jay Lilian Kneip
TL;DR
The paper advances a unified theory of representing abstract separation systems by concrete combinatorial structures, notably graphs, sets, and set bipartitions. It identifies precise structural conditions—scrupulousness for set representations, distributivity combined with scrupulousness for strong set representations, and fastidiousness for bipartition representations—under which a given system or universe can be realized, and it provides constructive methods and counterexamples to illustrate these criteria. A key contribution is the graphic-implementation criterion: finite distributive universes whose small separations form a boolean algebra with a degenerate maximal element admit a graphic realization, with an explicit construction of a graph $G$ such that $\mathcal{U}(G)$ is isomorphic to the given universe. The results illuminate the relationships between abstract separation axioms and concrete representations, enabling applications to tangles, tree-width duality, and related areas, while also clarifying when certain representations are impossible.
Abstract
Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.