Boolean building set arrangements

Abstract

In this article, we generalize the notion of connected subgraph arrangements, established by Cuntz and K\"uhne by allowing edges of cardinality greater than two, with a focus on local properties. We show that the class of free connected hypersubgraph arrangements is larger than the class of free connected subgraph arrangements, but that there exists a necessary condition for freeness that refers back to the graphic case. Furthermore, we obtain a unique description of these arrangements by associating them with Boolean building sets. We study the nested complex of the building set in relation to the lattice of intersections, and characterize all simplicial arrangements within this new class.