Absolutely closed semigroups

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