The flow lattice of oriented matroids

Recently Hochstättler and Nesetril introduced the flow lattice of an oriented matroid as generalization of the lattice of all integer flows of a digraph or more general a regular matroid. This lattice is defined as the integer hull of the characteristic vectors of signed circuits. Here, we characterize the flow lattice of oriented matroids that are uniform or have rank 3 with a particular focus on the dimension of the lattice and construct a basis consisting of directed circuits. For general oriented matroids we introduce a 2-sum and decompose oriented matroids into 3-connected parts. We show how to determine the dimension of the lattice of 2-sums and conclude with some questions based on extensive experiments on small oriented matroids with connectivity at least 3