The construction of all the star operations and all the semistar operations on 1-dimensional Prüfer domains

Abstract

Let Σ(D) (resp., Σ′(D)) be the set of star (resp., semistar) operations on a domain D. E.Houston gave necessary and sufficient conditions for an integrally closed domain D to have |Σ(D)| < ∞. Moreover, under those conditions, he gave the cardinality |Σ(D)| (Booklet of Abstracts of Conference: Commutative Rings and their Modules, 2012, Bressanone, Italy). We proved that an integrally closed domain D has |Σ′(D)| < ∞ if and only if it is a finite dimensional Prüfer domain with finitely many maximal ideals. Also we gave conditions for a pseudo-valuation domain (resp., an almost pseudo-valuation domain) D to have |Σ′(D)| < ∞. In this paper, we study star and semistar operations on a 1-dimensional Prüfer domain D. We aim to construct all the star and semistar operations on D. We introduce a sigma operation on D, and show that every semistar operation on D is expressed as a unique product of a star operation and a sigma operation.