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 K[S]. We define the type of S, type(S), in terms of some Apery sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K[S] is Cohen-Macaulay. If K[S] is a d-dimensional Cohen-Macaulay ring of embedding dimension at most d + 2, then type(S) <= 2. Otherwise, type(S) 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. (C) 2021 Elsevier B.V. All rights reserved.