  This paper investigates the projective closure of simplicial affine
semigroups in $\mathbb{N}^{d}$, $d \geq 2$. 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 $\mathbb{N}^d$, and
investigate its relationship with the original simplicial affine semigroup.
