A directed hypergraph GG consists of a vertex set VV along with a collection of directed hyperedges (A,B)(A,B), where AA and BB are finite subsets of VV. Given a set of vertices XX, we think of the edge (A,B)(A,B) as being on the boundary of XX if XX intersects AA and does not completely contain BB.
We can generalize the notion of directed hypergraph as follows. A filter graph GG consists of an infinite vertex set VV along with a collection of edges (F,G)(F,G), where FF and GG are filters on VV. Given a set of vertices XX, we think of the edge (F,G)(F,G) as being on the boundary of XX if XX is FF-positive and the complement of XX is GG-positive.
Filter graphs seem to be surprisingly graph-like. We'll show that filter graphs satisfy the natural generalization of the max-flow/min-cut theorem, where point masses flowing along directed edges in the usual hypergraph setting are replaced by ultrafilters flowing along filter-edges.