  Let $G$ be a connected reductive group over a perfect field $k$. We study a
certain normal reductive monoid $\overline M$ associated to a parabolic
$k$-subgroup $P$ of $G$. The group of units of $\overline M$ is the Levi factor
$M$ of $P$. We show that $\overline M$ is a retract of the affine closure of
the quasi-affine variety $G/U(P)$. Fixing a parabolic $P^-$ opposite to $P$, we
prove that the affine closure of $G/U(P)$ is a retract of the affine closure of
the boundary degeneration $(G \times G)/(P \times_M P^-)$. Using idempotents,
we relate $\overline M$ to the Vinberg semigroup of $G$. The monoid $\overline
M$ is used implicitly in the study of stratifications of Drinfeld's
compactifications of the moduli stacks $\mathrm{Bun}_P$ and $\mathrm{Bun}_G$.
