Projective Closure of Semigroup Algebras

This paper investigates the projective closure of simplicial affine semigroups in Nᵈ, d ≥ 2. We present a characterization of the Cohen-Macaulay property for the projective closure of these semigroups using Gr\"{o}bner bases. Additionally, we establish a criterion, based on Gr\"{o}bner bases, for determining the Buchsbaum property of non-Cohen-Macaulay projective closures of numerical semigroup rings. Lastly, we introduce the concept of k-lifting for simplicial affine semigroups in Nᵈ, and investigate its relationship with the original simplicial affine semigroup.