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 ( \(\textrm{MPD}\) $\text{MPD}$ ) are not Cohen–Macaulay, but they may be Buchsbaum. We give a necessary and sufficient condition for simplicial \(\textrm{MPD}\) $\text{MPD}$ -semigroups to be Buchsbaum in terms of pseudo-Frobenius elements. We give certain characterizations of \(\prec \) $\prec$ -almost symmetric \({\mathcal {C}}\) $\mathcal{C}$ -semigroups. When the cone is full, we prove the irreducible \({\mathcal {C}}\) $\mathcal{C}$ -semigroups, and \(\prec \) $\prec$ -almost symmetric \({\mathcal {C}}\) $\mathcal{C}$ -semigroups with Betti-type three satisfy the extended Wilf conjecture. For \(e \ge 4\) $e\ge 4$ , we give a class of MPD-semigroups in \({\mathbb {N}}^2\) ${\mathbb{N}}^2$ 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 \({\mathbb {N}}^d\) ${\mathbb{N}}^d$ , which satisfy the Arf property, and prove that Arf submonoids containing multiplicity are \(\textrm{PI}\) $\text{PI}$ -monoids.