Abstract

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.

Highlights

  • Hrushovski introduces the concept of a matroid being flat in [8] in order to prove the existence of a non trivial strongly minimal set that does not interpret an infinite field

  • We show examples of gammoids with each possible flatness degree

  • In [5], Evans shows that for finite matroids the notion of flatness characterizes the matroids known as strict gammoids, a class of matroids that arises from directed graphs

Read more

Summary

Introduction

Hrushovski introduces the concept of a matroid being flat in [8] in order to prove the existence of a non trivial strongly minimal set that does not interpret an infinite field. In [5], Evans shows that for finite matroids the notion of flatness characterizes the matroids known as strict gammoids, a class of matroids that arises from directed graphs. The restrictions of a strict gammoid, are known as gammoids and they form a complete class of matroids that has been widely studied [3, 5, 9, 11, 12, 13, 14]. In [5], Evans shows that strict gammoids have a pseudomodular lattice of flats. We construct a gammoid showing that the converse is not true, that is being pseudomodular and having flatness degree 2.

Matroid theory background
Flatness Degree
Problems

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.