Integrally closed ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a proper ideal of \(S\). It is said that \(E\) is integrally closed if it is of the form \(S(s) = (s + \mathbb{N}) \cap S\) for some \(s \in S\).
It can be proven that \(E\) is integrally closed if, and only if, \(E = S(m(E))\), where \(m(E)\) denotes the multiplicity ideal of \(E\). On the other hand, a numerical semigroup \(S\) is an Arf semigroup if, and only if, every integrally closed ideal is stable.
Examples
\(\circ\) Let \(S\) a numerical semigroup, then the maximal ideal \(M = S^*\) is integrally closed, since \(S^* = (m(S) + \mathbb{N}) \cap S\), where \(m(S)\) denotes the multiplicity of \(S\) as a numerical semigroup.
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\), \(I = \{9, 10, 12, 14\}\) and \(E = I + S\), let us prove the equality \(E = (9 + \mathbb{N}) \cap S\). First, \((9 + \mathbb{N}) \cap S = \{9, 10, 12, 14, \rightarrow\}\), then we have to prove that \(E = \{9, 10, 12, 14, \rightarrow\}\).
Let \(e \in E\), then there exists \(i \in I\) and \(s \in S\) such that \(e = i + s\). If \(s = 0\), \(e \in I \subseteq \{9, 10, 12, 14, \rightarrow\}\), otherwise, \(e = i + s \ge i + 5 \ge 14\) and \(e \in \{9, 10, 12, 14, \rightarrow\}\).
From the fact that \(\{14, 15, 16, 17, 18\} \subseteq E\), it is deduced that for all \(n \ge 14\) it holds \(n \in E\) and \(\{9, 10, 12, 14, \rightarrow\} \subseteq E\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 15, 29, 31 \rangle\), \(I = \{72, 75, 77\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(13, 15, 29, 31);
<Numerical semigroup with 4 generators>
gap> I := [42, 43, 44, 45, 46, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 93];
[ 42, 43, 44, 45, 46, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 93 ]
gap> IS := I + S;
<Ideal of numerical semigroup>There is no direct function that determines whether a proper ideal is integrally closed or not, but it is easy to compute it from the fact that \(IS\) is integrally closed if, and only if, \(IS = (m(IS) + \mathbb{N}) \cap S\).
gap> m := Minimum(IS);
73
gap> N1 := NumericalSemigroupBySmallElements([0, m]);
<Numerical semigroup>
gap> L := MaximalIdeal(Intersection(N1,S));
<Ideal of numerical semigroup>
gap> IsSubset(IS, L) = IsSubset(L, IS);
trueFirst, the function Minimum returns the multiplicity of a proper ideal, \(m(IS)\); afterwards, the function NumericalSemigroupBySmallElements generates the numerical semigroup given a list, considered as the list of left elements along with the conductor of a numerical semigroup. Then, \(N1 = \{0, m(IS), \rightarrow\} = (m(IS) + \mathbb{N})\cup\{0\}\); also, the function Intersection computes the intersection of two given numerical semigroups. Therefore, \(N1 \cap S = ((m(IS) + \mathbb{N}) \cap S) \cup \{0\}\); then, the function MaximalIdeal returns the maximal ideal of the numerical semigroup, \(L = (m(IS) + \mathbb{N}) \cap S\); finally, given two subsets \(I,J\) of \(\mathbb{N}\), the function IsSubset returns true if \(J \subseteq I\). To sum up, \(IS = (73 + \mathbb{N}) \cap S\) and \(IS\) is an integrally closed ideal.
References
https://gap-packages.github.io/ numericalsgps.