  We present a study of semigroup compactifications of a semitopological
semigroup $S$ using certain filters on $S$. We characterize closed
subsemigroups and closed left, right, and two-sided ideals in any semigroup
compactification of any semitopological semigroup $S$ in terms of these filters
and in terms of ideals of the corresponding $m$-admissible subalgebra of
$C(S)$. Furthermore, we characterize those points in any semigroup
compactification of $S$ which belong either to the smallest ideal of the
semigroup compactification or to the closure of this smallest ideal.
