Bulletin of the
Korean Mathematical Society BKMS

The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional Pr\"ufer 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 Pr\"ufer domain with $Y$-graph spectrum, that is, $R$ is a Pr\"ufer domain with exactly two maximal ideals $M$ and $N$ and $Spec(R)=\{(0)\subsetneq P_{1}\subsetneq\dots\subsetneq P_{n-1}\subsetneq M, N\,|\, P_{n-1}\subsetneq N\}$. We also characterize non-local Pr\"ufer domains $R$ such that $|SStar(R)|=7$, respectively $|SStar(R)|=14$.

Keywords: star operation, semistar operation, Pr\"{u}fer domain, $Y$-graph spectrum

MSC numbers: 13F05, 13A15

Supported by: The author was supported by KFUPM under research project RG 1413