Type and Conductor of Simplicial Affine Semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring \(\mathbb{K}[S]\). We define the type of \(S\), \(\operatorname{type}\), in terms of some Apéry sets of \(S\) and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when \(\mathbb{K}[S]\) is Cohen-Macaulay. If \(\mathbb{K}[S]\) is a \(d\)-dimensional Cohen-Macaulay ring of embedding dimension at most \(d+2\), then \(\operatorname{type}\leq 2\). Otherwise, \(\operatorname{type}\) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of \(S\) as an ideal of its normalization.