  Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is
called (1) $\mathcal C$-$closed$ if $X$ is closed in every topological
semigroup $Y\in\mathcal C$ containing $X$ as a discrete subsemigroup, (2)
$ideally$ $\mathcal C$-$closed$ if for any ideal $I$ in $X$ the quotient
semigroup $X/I$ is $\mathcal C$-closed; (3) $absolutely$ $\mathcal C$-$closed$
if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$,
the image $h[X]$ is closed in $Y$, (4) $injectively$ $\mathcal C$-$closed$
(resp. $\mathcal C$-$discrete$) if for any injective homomorphism $h:X\to Y$ to
a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed (resp.
discrete) in $Y$. Let $\mathsf{T_{\!z}S}$ be the class of Tychonoff
zero-dimensional topological semigroups. For a semigroup $X$ let $V\!E(X)$ be
the set of all viable idempotents of $X$, i.e., idempotents $e$ such that the
complement $X\setminus\frac{H_e}e$ of the set $\frac{H_e}e=\{x\in X:xe=ex\in
H_e\}$ is an ideal in $X$. We prove the following results: (i) for any ideally
$\mathsf{T_{\!z}S}$-closed semigroup $X$ each subgroup of the center
$Z(X)=\{z\in X:\forall x\in X\;\;(xz=zx)\}$ is bounded; (ii) for any
$\mathsf{T_{\!z}S}$-closed semigroup $X$, each subgroup of the ideal center
$I\!Z(X)=\{z\in Z(X):zX\subseteq Z(X)\}$ is bounded; (iii) for any
$\mathsf{T_{\!z}S}$-discrete or injectively $\mathsf{T_{\!z}S}$-closed
semigroup $X$, every subgroup of $Z(X)$ is finite, (iv) for any viable
idempotent $e$ in an ideally (and absolutely) $\mathsf{T_{\!z}S}$-closed
semigroup $X$, the maximal subgroup $H_e$ is ideally (and absolutely)
$\mathsf{T_{\!z}S}$-closed and has bounded (and finite) center $Z(H_e)$.
