Star Operations on Numerical Semigroups

Abstract

It is proved that the number of numerical semigroups with a fixed number n of star operations is finite if n > 1. The result is then extended to the class of analytically irreducible residually rational one-dimensional Noetherian rings with finite residue field and integral closure equal to a fixed discrete valuation domain.

Star operations on numerical semigroups: antichains and explicit results

“…In analogy with [14] and [15], we shall follow the notation of [1]. For further informations about numerical semigroups, the interested reader may consult [13].…”

“…More generally, it has been studied when the set of star operations is finite [6]. This paper follows the approach of the previous papers [14] and [15], where the main problem studied was to find ways to estimate (or, if possible, to count precisely) the number of star operations on an arbitrary numerical semigroup, and to deterimine explicitly all the numerical semigroups with exactly n star operations. More specifically, in [14] it was proved that, if n > 1, then there are only a finite number of numerical semigroups with exactly n star operations, while [15] provided an explicit formula for the cardinality of the set of star operations on S when S has multiplicity 3.…”

“…The goal of this paper is to improve the estimates proved in [14], with the dual objective to obtain an asymptotic bound for the number of nonsymmetric numerical semigroups with n or less star operations and to determine explicitly such semigroups in the case n = 10. This is accomplished by studying a natural order on the set of nondivisorial ideals (introduced and sketched in [15]) and linking star operations with the antichains of this order; this allows to establish several inequalities between the size of Star(S) and the invariants of S.…”

Star Operations on Kunz Domains

“…In this context, a rich source of examples are semigroup rings, that is, subrings of the power series ring K[[X]] (where K is a field, usually finite) of the form K[[S]] := K[[X S ]] := { i a i X i | a i = 0 for all i / ∈ S}, where S is a numerical semigroup (i.e., a submonoid S ⊆ N such that N \ S is finite). Star operations can also be defined on numerical semigroups [13], and there is a link between star operations on S and star operations on K[[S]]: for example, every star operation on S induces a star operation on K[[S]], and |Star(S)| = 1 if and only if |Star(K[[S]])| = 1 [13,Theorem 5.3], with the latter result corresponding to the equivalence between S being symmetric and K[[S]] being Gorenstein [2,10]. A detailed study of star operations on some numerical semigroup rings was carried out in [14].…”

numericalsgps, a GAP package for numerical semigroups

“…We give a function to test if a numerical semigroup is almost-symmetric and include the procedure presented in [41] to compute the set of almost symmetric numerical semigroups with fixed Frobenius number. We provide functions for computing the small elements of an ideal (the definition is analogous to that in numerical semigroups), Apéry sets (and tables; see [20]), the ambient numerical semigroup, membership, and also some basic operations as addition, union, subtraction (I − J = {z ∈ Z | z + J ⊆ I}), set difference, multiplication by an integer, translation by an integer, intersection, blow-up ( n∈N nI − nI) and * -closure with respect to a family of ideals ( [44]). …”