Characterizing categorically closed commutative semigroups

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