Base-sortable Matroids and Koszulness of Semigroup Rings

Motivated by a question in commutative algebra and inspired by the work of Sturmfels, we introduce the class of base-sortable matroids and show that it is closed under several matroid operations. All matroids of rank 2 are base-sortable and we give a characterization of base-sortability by excluded minors in the case of graphic matroids and rank 3 matroids. Transversal matroids with certain presentations are also base-sortable. For a base-sortable matroid M, the basis monomial ring RMis shown to be Koszul, by proving that the toric ideal of this ring has a quadratic Grِbner basis. Extending the concept of combinatorial pure subrings considered by Herzog, Hibi and Ohsugi we define the matroid operations deletion, contraction and duality for homogeneous semigroup rings with square-free monomial generators, and observe that Koszulness is preserved under these operations.