Affine semigroups of maximal projective dimension-II

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 ≺-almost symmetric C-semigroups. When the cone is full, we prove the irreducible C-semigroups, and ≺-almost symmetric C-semigroups with Betti-type three satisfy the extended Wilf conjecture. For e≥4, 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 N^d, which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are PI-monoids.