On the NBC-complexes and ββ-invariants of abstract convex geometries

An (abstract) convex geometry is a combinatorial abstraction of convexity which is a Moore family with the closure operator satisfying the anti-exchange property. A number of results of matroids on the NBC-complexes (or broken circuit complexes) happen to have some exact analogues in convex geometries: for instance, the Whitney-Rota’s formula of the characteristic function of a matroid, Brylawski’s decomposition of the NBC-complexes, etc. A ββ-invariant of a convex geometry is derived from the characteristic function in the same way as that of a matroid. We introduce a merging of two convex geometries, which is called a 1-sum, and exhibit the resultant value of the ββ-invariant of a 1-sum.