Absolutely closed semigroups

Let $$ be a class of topological semigroups. A semigroup X is called absolutely $$-closed if for any homomorphism $$ to a topological semigroup $$, the image h[X] is closed in Y. Let $$, $$, and $$ be the classes of $$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely $$-closed if and only if X is absolutely $$-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 $$-closed if and only if X is finite. Also, for a given absolutely $$-closed semigroup X we detect absolutely $$-closed subsemigroups in the center of X.