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.
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