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.
Keywords: -closed semigroup; Chain-finite semigroup; Commutative semigroup; Group; Periodic semigroup; Semilattice.
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