Semistar operations on almost pseudo-valuation domains (Q2920212)

Let \(D\) be an integral domain, and let \(\Sigma' (D)\) be the set of semistar operations on it. The interesting question by the author is: ``When is the cardinality \(|\Sigma' (D)|\) finite?'' If \(D\) is an integrally closed domain, then it was proven by author \(| \Sigma' (D)| < \infty\) if and only if \(D\) is a finite dimensional Prüfer domain with only finite number of maximal ideals, see the book [Commutative Semigroup Rings. 2nd. ed. Kaisei, Tokyo, (2006)]. For a pseudo-valuation domain \(D\), the characterization of \(|\Sigma' (D)| < \infty\) was given by author in a published paper [JP J. Algebra Number Theory Appl. 17, No. 2, 163--172 (2010; Zbl 1198.13005)]. In this paper, the author studies star operations and semistar operations on an almost pseudo-valuation domain \(D\), and gives a necessary and sufficient condition for \(|\Sigma' (D)| < \infty\).