Let ▫$\mathcal{C}$▫ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup ▫$X$▫ is called ▫$\mathcal{C}$▫-▫$closed$▫ if ...▫$X$▫ is closed in each topological semigroup ▫$Y\in \mathcal{C}$▫ containing ▫$X$▫ as a discrete subsemigroup; ▫$X$▫ is ▫$projectively$▫ ▫$\mathcal{C}$▫-▫$closed$▫ if for each congruence ▫$\approx$▫ on ▫$X$▫ the quotient semigroup ▫$X/_\approx$▫ is ▫$\mathcal{C}$▫-closed. A semigroup ▫$X$▫ is called ▫$chain$▫-▫$finite$▫ if for any infinite set ▫$I\subseteq X$▫ there are elements ▫$x,y\in I$▫ such that ▫$xy \notin \{x,y\}$▫. We prove that a semigroup ▫$X$▫ is ▫$\mathcal{C}$▫-closed if it admits a homomorphism ▫$h:X\to E$▫ to a chain-finite semilattice ▫$E$▫ such that for every ▫$e\in E$▫ the semigroup ▫$h^{-1}(e)$▫ is ▫$\mathcal{C}$▫-closed. Applying this theorem, we prove that a commutative semigroup ▫$X$▫ is ▫$\mathcal{C}$▫-closed if and only if ▫$X$▫ is periodic, chain-finite, all subgroups of ▫$X$▫ are bounded, and for any infinite set ▫$A\subseteq X$▫ the product ▫$AA$▫ is not a singleton. A commutative semigroup ▫$X$▫ is projectively ▫$\mathcal{C}$▫-closed if and only if ▫$X$▫ is chain-finite, all subgroups of ▫$X$▫ are bounded and the union ▫$H(X)$▫ of all subgroups in ▫$X$▫ has finite complement ▫$X\setminus H(X)$▫.
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