  We study the set of star operations on local Noetherian domains $D$ of
dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral
closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to
the study of a class of closure operations (more precisely, multiplicative
operations) in a finite extension $k\subseteq B$, where $k$ is a field, and
then we study how the cardinality of this set of closures vary as the size of
$k$ varies while the structure of $B$ remains fixed.
