Star operations on overrings of Prüfer domains

Let R be a Prüfer domain. We begin with two questions: (Q1) If R has only finitely many star operations, is the same true for each overring of R? and (Q2) If each proper overring of R has only finitely many star operations, is the same true for R? We show that both questions have negative answers in general, we give a complete description of when (Q1) has a positive answer (loosely, R must be “almost” strongly discrete), and we show that for a finite dimensional Prüfer domain, the answer to (Q2) is always “yes”. We then study star regular Prüfer domains, that is, Prüfer domains R such that |Star(T)|≤|Star(R)| for each overring T of R, and we compute the number of star operations on some particular Prüfer domains.