On The Cardinality of Star and Semistar Operations of some Classes of Integral Domains

In this project, we will study the cardinality of star and semistar operations of some classes of integral domains. Our concern is to gave answers to the following questions: given a Prufer domainR, (1) If $R$ has an overring $T$ such that $Star(T)$ is infinite is $Star(R)$ infinite? (2) If each proper overring of $R$ has only finitely many star operations, does $R$ itself have only finitely many star operations? Also we will study some classes of Prufer domains with only finitely many semistar operations in terms of their prime spectrum. Namely, we deal with n-dimensional Prufer domains $R$ with $n\geq 2$ such that $|SStar(R)| = |Star(R)| + |Spec(R)| + |\Omega$ where $\Omega$ is the set of all nonzero prime idempotent ideals of $R$ and the number of semistar operation of a one-dimensional nonlocal Prufer domain with $Y$-graph spectrum.