An implicational base is a set of rules: “if these attributes hold, then this attribute must also hold.” Every closure system — a family of sets closed under intersection — has an implicational base, but the base is far from unique. Finding the base with the minimum total size (fewest premises and conclusions combined) is NP-hard in general.
The paper identifies a structural condition that makes the optimization tractable for convex geometries — closure systems where every closed set can be built by adding one element at a time, each element being an “extreme point” of the set it's added to. Convex geometries arise from convex hulls, from graph search orderings, from lattice theory.
The key tool: quasi-closed hypergraphs. A quasi-closed set is almost closed — its closure adds exactly one new element. These sets index the essential implications, and the hypergraph structure (which quasi-closed sets overlap, which are disjoint) determines the optimization complexity.
The result: when the quasi-closed hypergraph has disjoint edges — no quasi-closed set shares elements with another — the optimal base is computable in polynomial time. Disjointness means the implications don't interact: optimizing one implication doesn't affect the others, so the global optimum is the collection of local optima.
This unifies several previously known tractability results — double-shelling, acyclic, affine, acceptant convex geometries — all turn out to satisfy the disjoint-edges condition. The individual results were proved separately; the hypergraph condition reveals they're all instances of the same phenomenon.