Conductor ideal
Definition
Let \(S\) be a numerical semigroup. It is defined the conductor ideal of \(S\) as the set \(S - \mathbb{N} = \{z \in \mathbb{Z} ~ | ~ z + \mathbb{N} \subseteq S \}\). Equivalently, the conductor ideal of \(S\) is defined as the numerical semigroup \(S - \mathbb{N} = \{C(S), \rightarrow \}\), where \(C(S)\) denotes the conductor of \(S\).
Examples
\(\circ\) Let \(S = \langle 5, 8, 11 \rangle\). Computing the first elements of \(S\), it is obtained that \(S = \{0, 5, 8, 10, 11, 13, 15, 16, 18, \rightarrow \}\), thus \(C(S) = 18\) and \(S - \mathbb{N} = \{18, \rightarrow \}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 15, 20 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 15, 20);
<Numerical semigroup with 3 generators>Given a numerical semigroup, the function Conductor returns the conductor of the numerical semigroup.
gap> C = Conductor(S);
78Then, the conductor ideal of \(S\) is \(S - \mathbb{N} = \{78, \rightarrow\}\). The minimal system of generators of the conductor ideal is \(I = \{C(S), C(S) + 1, \ldots, C(S) + m(S)\}\), where \(m(S)\) denotes the multiplicity of \(S\). The function Multiplicity computes the multiplicity of a given numerical semigroup.
gap> M := Multiplicity(S);
13
gap> IS := [C..C+M] + S;
<Ideal of numerical semigroup>\(\diamond\) Given a relative ideal \(IS\) of \(S\), it can be defined the conductor of \(IS\) as the least element \(x \in IS\) such that \(s + \mathbb{N} \subseteq IS\). This element exists since \(IS + S \subseteq IS\). The functions Conductor and ConductorOfIdealOfNumericalSemigroup compute the conductor of a given relative ideal.
gap> S := NumericalSemigroup(33, 49, 76, 133);
<Numerical semigroup with 4 generators>
gap> IS := [-13, -5, 6, 21] + S;
<Ideal of numerical semigroup>
gap> Conductor(IS);
232References
https://gap-packages.github.io/ numericalsgps.