Canonical module of Cohen-Macaulay affine semigroup

Let S be a simplicial affine semigroup of dimension r and R=K[[S]] be the semigroup ring assigned to S, where K is a field. Then R is a Noetherian ring of krull dimension r. When r=1, S is a numerical semigroup whose assigned semigroup ring is a one dimensional Cohen-Macaulay ring. In this case, all each set of S is a finite set, and the number of its maximal elements with respect to the natural relation, is equal to the type of R. But in general, when r>1, the Apery sets of S are not necessarily finite. In this paper, we introduce r Apery sets of S whose intersection is a finite set and determines the type and the canonical module of R. This set coincides with an Apery set, when r=1. In particular, we extend the known facts about canonical module of numerical semigroups to all Cohen-Macaulay simplicial affine semigroups.