Absolutely closed semigroups.

Let $C$ be a class of topological semigroups. A semigroup X is called absolutely $C$-closed if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in C$, the image h[X] is closed in Y. Let $T_{1}S$, $T_{2}S$, and $T_{z}S$ be the classes of $T_1$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely $T_{z}S$-closed if and only if X is absolutely $T_{2}S$-closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely $T_{1}S$-closed if and only if X is finite. Also, for a given absolutely $C$-closed semigroup X we detect absolutely $C$-closed subsemigroups in the center of X.