The semistar operations on certain Prüfer domain, II

Let D be a 1-dimensional Prüfer domain with exactly two maximal ideals. We completely determine the star operations and the semistar operations on D. Let G be a torsion-free abelian additive group. If G is not discrete, G is called indiscrete. If every non-empty subset S of G which is bounded below has its infimum inf(S) in G, then G is called complete. If G is not complete, G is called incomplete. Let D be a 1-dimensional Prüfer domain with exactly two maximal ideals M and N , let V = DM (resp., W = DN ), let v (resp., w) be a valuation belonging to V (resp., W ), let Γ (resp., Γ′) be the value group of v (resp., w), and let K be the quotient field of D. Let Σ(D) (resp., Σ′(D)) be the set of star operations (resp., semistar operations) on D. In [1, Proposition 6 (2)], we proved that, if both Γ and Γ′ are discrete, then |Σ(D)| = 1 and |Σ′(D)| = 7. In this paper, we prove the following, Theorem Let D be a 1-dimensional Prüfer domain with exactly two maximal ideals M and N , let Γ (resp., Γ′) be the value group belonging to the valuation ring DM (resp., DN ). (1) If both Γ and Γ′ are discrete, then |Σ(D)| = 1 and |Σ′(D)| = 7. (2) If Γ is discrete, and if Γ′ is indiscrete, then |Σ(D)| = 2 and |Σ′(D)| = 14. (3) If both Γ and Γ′ are indiscrete, then |Σ(D)| = 4 and |Σ′(D)| = 30. In the above Theorem, (1) was proved in [1, Proposition 6 (2)] as was mentioned. (2) was proved in [2] in the case where Γ′ is incomplete. (3) was studied in [2] in the case where both Γ and Γ′ are incomplete. Unfortunately, [2, Theorem (4)] and its proof were incomplete. Section 1 of this paper contains preliminary results, and in Section 2, we prove Theorem (3). The proof of Theorem (2) is similar to that of [2, Theorem (2)]. Throughout the paper, we confer [2], p denotes an element of M \N , q denotes an element of N \M , and x denotes an element of K \ {0}. Received 8 August 2013; revised 24 October 2013 2000 Mathematics Subject Classification. Primary 13A15