Stable proper ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a proper ideal of \(S\) with reduction number \(r(E)\). It is said that \(E\) is stable if \(r(E) = 1\), that is, if \(2E = m(E) + E\), where \(nE = \{e_1 + e_2 + \cdots + e_n ~ | ~ e_1, \ldots, e_n \in E\}\) for all \(n \in \mathbb{N} \setminus \{0\}\) and \(m(E)\) denotes the multiplicity of \(E\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle\), \(I = \{2, 4\}\) and \(E = I + S = \{2, 4, 5, 7, \rightarrow \}\). For \(h = 1\), since \(m(E) = 2\),
\[ 2E = E + E = \{4, 6, \rightarrow\} \ne \{4, 6, 7, 9, \rightarrow \} = 2 + E. \]
Therefore, \(E\) is not stable. It is easy to prove that \(r(E) = 2\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 12, 15, 19, 20 \rangle\), \(I = \{24, 38, 40\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(12, 15, 19, 20);
<Numerical semigroup with 4 generators>
gap> I := [24, 38, 40];
[ 24, 38, 40 ]
gap> IS := I + S;
<Ideal of numerical semigroup>There is no direct method that states if a proper ideal is stable or not, but the function Reduction number returns the reduction number of a proper ideal.
gap> ReductionNumber(IS);
1It is obtained that \(IS\) is a stable proper ideal.
References
https://gap-packages.github.io/ numericalsgps.