The Structure of Submodular Separation Systems

TL;DR

This work analyzes the structure and submodularity of abstract separation systems, distinguishing external (structure-based) submodularity from order-induced submodularity and showing how to realize the former within a universe via the Dedekind-MacNeille completion. It proves that every submodular separation system is submodular in some universe, but provides counterexamples demonstrating that submodularity in a universe need not arise from a submodular order function. It then develops methods to extend witnessing submodular functions and to decompose distributive-universe separation systems into simpler corner-closed components, with a key emphasis on corner-faithful embeddings into bipartition universes through a distributive-Birkhoff framework. These results unify and extend the tangle-theory toolbox, offering explicit decomposition schemes and embedding techniques that clarify the relationship between internal and external notions of submodularity and enable broader applicability to graphs, matroids, and related combinatorial structures.

Abstract

We analyse various structural and order-theoretical aspects of abstract separation systems and partial lattices, as well as the relationship between the different submodularity conditions one can impose on them.

Figures (3)

• Figure 1: The Hasse diagram of $\vU$ from \ref{['thm:example_bip']}. For readability, only points in $\vS$ are labelled and only one side of each bipartition is denoted.
• Figure 2: The dark blue elements form a partial lattice, which does not contain a cycle in the inner dependency digraph, however the green dashed edges form a cycle in the outer dependency digraph
• Figure 3: The dark blue elements form a submodular partial lattice which, however it is embedded into a lattice, has the cycle indicated in green in its dependency digraph.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 2.1: Birkhoff representation theorem; cf. LatticeBook*§ 5.12
• Theorem 3.1
• proof : Proof of \ref{['thm:submodular']}.
• Theorem 3
• Lemma 4.1
• proof
• Corollary 4.2