Abstract
Let C be a class of topological semigroups. A semigroup X is called absolutely C-closed if for any homomorphism h:X → Y to a topological semigroup Y ∈ C, the image h[X] is closed in Y. Let T 1S, T 2S, and T zS be the classes of T1, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely T zS-closed if and only if X is absolutely T 2S-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 1S-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.