THE NUMBER OF STAR OPERATIONS ON NUMERICAL SEMIGROUPS AND ON RELATED INTEGRAL DOMAINS DARIO SPIRITO Abstract. We study the cardinality of the set Star(S) of star op- erations of a numerical semigroup S; in particular, we study ways to estimate Star(S) and to bound the number of nonsymmetric numerical semigroups such that jStar(S)j ≤ n. We also study this problem in the setting of analytically irreducible, residually ratio- nal rings whose integral closure is a fixed discrete valuation ring. 1. Introduction A star operation on an integral domain D is a particular closure op- eration on the set of fractional ideals of D; this notion was defined to generalize the divisorial closure [13, 5] and has been further general- ized to the notion of semistar operation [16]. Star operations have also been defined on cancellative semigroups in order to obtain semigroup- theoretic analogues of some ring-theoretic (multiplicative) definitions [11]. A classical result characterizes the Noetherian domains D in which every ideal is divisorial or, equivalently, which Noetherian domains ad- mit only one star operation: this happens if and only if D is Gorenstein of dimension one [2]. Recently, this result has been a starting point of a deeper investigation on the cardinality of the set Star(D) of the star op- erations on D, obtaining a precise counting for h-local Pr¨uferdomains [7] (and, more generally, an algorithm to calculate their number for semilocal Pr¨uferdomains [25]), some pseudo-valuation domains [17, 24] and some Noetherian one-dimensional domains [8, 9, 23]. In particu- lar, for Noetherian domains, a rich source of examples are numerical semigroup rings, that is, rings in the form K[[S]] := K[[Xs j s 2 S]], where K is a field and S is a numerical semigroup. Inspired by this example, the study of star operations on numerical semigroups (and, in particular, of their cardinality) was initiated in [20]. In particular, the main problem that was tackled was the follow- ing: given a (fixed) integer n, how many numerical semigroups have exactly n star operations? By estimating the cardinality of Star(S), it was shown that this number is always finite, and that the same holds 2010 Mathematics Subject Classification. 20M12; 13G05. Key words and phrases. Numerical semigroups; Star operations; Residually ra- tional rings. 1 2 DARIO SPIRITO for residually rational rings (see Section 10 for a precise statement). Subsequently, in [26], better estimates on jStar(S)j allowed to give a much better bound the number of semigroups with at most n star op- erations, while in [21] the set Star(S) was described in a very precise way when the semigroup S has multiplicity 3. In this paper, we give a unified treatment of the study of Star(S), surveying the main results of [20], [21], [26] and [22] and deepening them. In particular, we give a rather precise asymptotic expression for the number of semigroups of multiplicity 3 with less than n star operations (Theorem 6.4), an O(n) bound for the semigroups of prime multiplicity (Theorem 7.4), we list all nonsymmetric numerical semi- groups with 150 or less star operations (Table 4), and prove an explicit bound for residually rational rings (Theorem 10.5). The structure of the paper is as follows: Sections 2 and 3 present basic material; Sections 4 and 5 present estimates already present in [20] and [26]; Section 6 deepens the analysis of [21] on semigroups of multiplicity 3; Section 7 studies the case where the multiplicity is prime (and bigger than 3); Section 8 introduces the concept of linear families (one example of which was analyzed in [22]); Section 9 is devoted to algorithms to calculate jStar(S)j and to determine all the nonsymmetric semigroups with at most n star operations; Section 10 studies the domain case, and contains analogues of the results of Section 4 for residually rational domains. 2. Notation For all unreferenced results on numerical semigroups we refer the reader to [19]. A numerical semigroup is a set S ⊆ N that contains 0, is closed by addition and such that N n S is finite. If a1; : : : ; an are coprime positive integers, the numerical semigroup generated by a1; : : : ; an is Pn ha1; : : : ; ani := f i=1 tiai j ti 2 Ng. The notation S = f0; b1; : : : ; bn; ! g indicates that S is the set containing 0; b1; : : : ; bn and all integers bigger than bn. To any numerical semigroup S are associated some natural numbers: • the genus of S is g(S) := jN n Sj; • the Frobenius number of S is F (S) := sup(N n S); • the multiplicity of S is m(S) := inf(S n f0g). A hole of S is an integer x 2 N n S such that F (S) − x2 = S.A semigroup S is symmetric if it has no holes, while it is pseudosymmetric if g(S) is even and g(S)=2 is its only hole. An integral ideal of S is a nonempty subset I ⊆ S such that I+S ⊆ I, i.e., such that i + s 2 I for all i 2 I, s 2 S.A fractional ideal of S is a subset I ⊆ Z such that d + I is an integral ideal for some d 2 Z, STAR OPERATIONS ON NUMERICAL SEMIGROUPS 3 or equivalently an I (Z such that I + S ⊆ I. We shall use the term \ideal" as a shorthand for \fractional ideal". If fIαgα2A is a family of ideals, then its intersection (if nonempty) is an ideal, while its union is an ideal if and only if there is a d 2 Z such that d < i for all i in the union. If I;J are ideals, the set (I − J) := fx 2 Z j x + J ⊆ Ig is still an ideal of S. We denote by F(S) the set of fractional ideals of S, and by F0(S) the set of fractional ideals contained between S and N; equivalently, F0(S) = fI 2 F(S) j 0 = inf(I)g. For every ideal I, there is a unique d such that −d + I 2 F0(S) (namely, d = inf(I)). If a; b are integers, we use (a; b) to indicate their greatest common divisor. If f; g are functions of n, we use f = O(g) to mean that there is a constant C such that f(n) ≤ C · g(n) for all n ≥ 0. 3. Star operations Definition 3.1. [20, Definition 3.1] A star operation is a map ∗ : F(S) −! F(S), I 7! I∗, that satisfies the following properties: •∗ is extensive: I ⊆ I∗; •∗ is order-preserving: if I ⊆ J, then I∗ ⊆ J ∗; •∗ is idempotent: (I∗)∗ = I∗; •∗ fixes S, that is, S∗ = S; •∗ is translation-invariant: d + I∗ = (d + I)∗. We denote by Star(S) the set of star operations on S. If I = I∗, we say that I is ∗-closed; we denote the set of ∗-closed ideals by F ∗(S). The set Star(S) can be endowed with a natural partial order: we ∗1 ∗2 say that ∗1 ≤ ∗2 if I ⊆ I for every ideal I, or equivalently if F ∗2 (S) ⊆ F ∗1 (S). Under this order, Star(S) is a complete lattice: its minimum is the identity, while its maximum is the v-operation (or divisorial closure) v : I 7! (S − (S − I)). Since N is v-closed, any star operation restricts to a map ∗0 : F0(S) −! F0(S); furthermore, ∗0 uniquely determines ∗ (since every ideal can be v translated into F0(S)). We define G0(S) := F0(S) n F (S), that is, v G0(S) is the set of ideals I of S such that 0 = inf I and I 6= I . Since F0(S) is finite, Star(S) is a finite set for all numerical semigroup S [20, Proposition 3.2]. Furthermore, jStar(S)j = 1 if and only if v is the identity, which happens if and only if S is symmetric [1, Proposition I.1.15]. 4. Estimates through the genus Our main interest in this paper will be the function Ξ(n) that asso- ciates to every integer n > 1 the number of numerical semigroups S such that 2 ≤ jStar(S)j ≤ n. More generally, if S is a set of numerical semigroups, we define ΞS (n) as the number of semigroups S 2 S such 4 DARIO SPIRITO that 2 ≤ jStar(S)j ≤ n. We will mainly be interested in the asymptotic growth and in asymptotic bounds of Ξ and ΞS , for some distinguished set S of semigroups. It is very difficult to determine precisely the number of star opera- tions on a numerical semigroup S, while it is easier to find estimates for jStar(S)j: for this reason, we work with Ξ instead of the function that counts the number of semigroups with exactly n star operations. Most of the bounds proven in the paper will be obtained in a two-step process: (1) find a function φ (depending on some of the invariants of S) such that jStar(S)j ≥ φ(S) for all S 2 S; (2) estimate the number of S 2 S satisfying φ(S) ≤ n. In this way, we obtain an estimate on the number of semigroups S 2 S satisfying jStar(S)j ≤ n: indeed, if jStar(S)j ≤ n then we must also have φ(S) ≤ n. The first important result is to prove that Ξ is actually well-defined, that is, that there are only a finite number of numerical semigroups satisfying 2 ≤ jStar(S)j ≤ n.