Conductor of an ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). By definition of relative ideal, it holds that \(E+S \subseteq E\), where
\[ E + S = \{e + s ~ | ~ e \in E, s \in S\}, \]
and then there exists \(c \in E\) such that \(c + \mathbb{N} \subseteq E\). It is defined the conductor of \(E\) as the least element satisfying the previous property, and it is denoted by \(C(I)\).
This definitions is an extension of the definition of conductor of a numerical semigroup, considering \(S\) as ideal of itself.
Examples
\(\circ\) Let \(S = \langle 6, 8, 13 \rangle = \{0, 6, 8, 12, 13, 14, 16, 18, 19, 20, 21, 22, 24, \rightarrow\}\) and \(I = \{-5, -3, -2\} + S\). Let \(L = \{0, 6, 8, 12, 13, 14, 16, 18, 19, 20, 21, 22\}\) and \(L - n = \{l - n ~ | ~ l \in l\}\), with few computations,
\[ \begin{align*} I &= (\{-5\} + S) \cup (\{-3\} + S) \cup (\{2\} + S) = (L - 5) \cup (L-3) \cup (L + 2) \cup \{19, \rightarrow\} \\ & = \{-5, -3, 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, \rightarrow \}, \end{align*} \]
concluding that \(C(I) = 13\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 14, 23, 43, 68, 81, 87 \rangle\) and \(I = \{-97, -85\} + S\), in GAP:
gap> S := NumericalSemigroup(14, 23, 43, 68, 81, 87);
<Numerical semigroup with 6 generators>
gap> I := [-97, -85] + S;
<Ideal of numerical semigroup>Given a relative ideal \(I\) of a numerical semigroup, the functions Conductor and ConductorOfIdealOfNumericalSemigroup return the conductor of \(I\).
gap> Conductor(I);
-6
gap> ConductorOfIdealOfNumericalSemigroup(I) = Conductor(I);
trueIf in function Conductor we give a numerical semigroup \(S\), then it computes the conductor of \(S\).
gap> Conductor(S);
91References
https://gap-packages.github.io/ numericalsgps.