On the associated graded ring of semigroup algebras

In this paper, we give a necessary and sufficient condition for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroup using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also discuss nice extensions for simplicial affine semigroups, a concept which is motivated by nice extensions of numerical semigroups.