The koszul property in affine semigroup rings

We investigate the Koszul property for quotients of affine semigroup rings by semigroup ideals. Using a combinatorial and topological interpretation for the Koszul property in this context, we recover known results asserting that certain of these rings are Koszul. In the process, we prove a stronger fact, suggesting a more general definition of Koszul rings, already considered by Fröberg. This more general definition of Koszulness turns out to be satisfied by all Cohen-Macaulay rings of minimal multiplicity.