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