The Cartesian product of hypergraphs

We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and erovnik for the case of graphs and introduces the notion of diagonal-free grids as a replacement of the chord-free 4-cycles that play a crucial role in the case of graphs. This leads to a generalization of relation on the arc set, whose convex hull is shown to coincide with the product relation of the prime factorization.