On the number of semistar operations of some classes of prÜfer domains

The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional Prüfer domains. We prove that |SStar(R)| = |Star(R)| + |Spec(R)| + |Idem(R)| where Idem(R) is the set of all nonzero idempotent prime ideals of R if and only if R is a Prüfer domain with Y-graph spectrum, that is, R is a Prüfer domain with exactly two maximal ideals M and N and Spec(R) = {(0) ⊊ P₁ ⊊ · · · ⊊ P_{n −1} ⊊ M, N | P_{n −1} ⊊ N}. We also characterize non-local Prüfer domains R such that |SStar(R)| = 7, respectively |SStar(R)| = 14.