  A semigroup $X$ is $absolutely$ (resp. $injectively$) $T_1S$-$closed$ if for
any (injective) homomorphism $h:X\to Y$ to a $T_1$ topological semigroup
$Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. We prove that a
commutative semigroup $X$ is injectively $T_1S$-closed if and only if $X$ is
bounded, nonsingular and Clifford-finite. Using this characterization, we prove
that (1) every injectively $T_1S$-closed semigroup has injectively
$T_1S$-closed center, and (2) every absolutely $T_1S$-closed semigroup has
finite center.
