Gammoids, Pseudomodularity and Flatness Degree

We introduce the concept of flatness degree for matroids, as a generalization of submodularity. This represents weaker variations of the concept of flatness which characterize strict gammoids for finite matroids. We prove that having flatness degree 3, which is the smallest non-trivial flatness degree, implies pseudomodularity on the lattice of flats of the matroid. We show however an example of a gammoid for which the converse is not true. We also show examples of gammoids with each possible flatness degree. All of this examples show that pseudomodular gammoids are not necessarily strict.