Affine Semigroups of Maximal Projective Dimension

A submonoid of N^d is of maximal projective dimension (MPD) if the associated affine semigroup k-algebra has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of MPDsemigroups in N^d. We prove that under suitable conditions these semigroups satisfy the generalizedWilf’s conjecture. We prove that the generic nature of the defining ideal of the associated semigroup algebra of an MPD-semigroup implies the uniqueness of the row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of MPD-semigroups. We prove that the defining ideal of gluing of MPD-semigroups is never generic.