  In this paper we search for conditions on a countably compact
(pseudo-compact) topological semigroup under which: (i) each maximal subgroup
$H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part
$H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a
closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to
H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$,
$\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$,
is continuous.
