Absolutely closed semigroups

Abstract

Let $${\mathcal {C}}$$ C be a class of topological semigroups. A semigroup X is called absolutely $${\mathcal {C}}$$ C -closed if for any homomorphism $$h:X\rightarrow Y$$ h : X → Y to a topological semigroup $$Y\in {\mathcal {C}}$$ Y ∈ C , the image h[X] is closed in Y. Let $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$ T 1 S , $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$ T 2 S , and $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$ T z S be the classes of $$T_1$$ T 1 , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely $$\textsf {T}_{\!\textsf {z}}\textsf {S}$$ T z S -closed if and only if X is absolutely $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$ 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 $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$ T 1 S -closed if and only if X is finite. Also, for a given absolutely $${\mathcal {C}}$$ C -closed semigroup X we detect absolutely $${\mathcal {C}}$$ C -closed subsemigroups in the center of X.