Affine semigroups of maximal projective dimension-II

Abstract

If the Krull dimension of the semigroup ring is greater than one, then affine semigroups of maximal projective dimension (MPD) are not Cohen–Macaulay, but they may be Buchsbaum. We give a necessary and sufficient condition for simplicial MPD -semigroups to be Buchsbaum in terms of pseudo-Frobenius elements. We give certain characterizations of α -almost symmetric C-semigroups with . When the cone is full, we prove the irreducible C -semigroups, and α-almost symmetric C-semigroups with Betti-type three satisfy the extended Wilf conjecture. For , we give a class of MPD-semigroups in N2 such that there is no upper bound on the Betti-type in terms of embedding dimension e. Thus, the Betti-type may not be a bounded function of the embedding dimension. We further explore the submonoids of Nd, which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are PI-monoids.