Conductor
Definition
Let \(S\) be a numerical semigroup. It is said that \(s \in S\) is the conductor of \(S\) if is the least integer which satisfies that for any \(n \in \mathbb{N}\), it holds \(s + n \in S\), and it is denoted by \(C(S)\) or \(\chi(S)\). Equivalently, \(s \in S\) is the conductor of \(S\) if \(s-1 \not \in S\) and for all \(n \in \mathbb{N}\) such that \(n \ge s\), it holds \(n \in S\).
There is a obvious relation between the conductor and the Frobenius number, that is, \(F(S) + 1 = C(S)\). The conductor does not determine in general the numerical semigroup, that means, it may exist more than one numerical semigroup with the same conductor.
Examples
\(\circ\) Let us consider a non-zero \(m \in \mathbb{N}\) arbitrary but fixed and \(S = \{0, m, \rightarrow\}\). Obviously, \(C(S) = m\) and in this case every element of \(S \setminus \{0\}\) is greater or equal than the conductor. This kind of numerical semigroups are the only ones that satisfy this property. This example shows that for any \(n \in \mathbb{N}\) with \(n \ne 1\), there exists a numerical semigroup \(S\) such that \(C(S) = n\).
\(\circ\) Let \(S_1 = \langle 5, 7, 9 \rangle\) and \(S_2 = \langle 2, 15 \rangle\). With simple calculations we obtain that \(S_1 = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(S_2 = \{0, 2,4,6,8,10,12,14, \rightarrow\}\), therefore \(C(S_1) = 14 = C(S_2)\) and \(S_1 \ne S_2\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 3, 7, 9, 11 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 7, 9, 11);
<Numerical semigroup with 4 generators>Using the Conductor function, we can obtain the conductor of \(S\).
gap> Conductor(S);
9Taking into account the equality \(F(S) + 1 = C(S)\), we can compute the conductor from the Frobenius number. The function FrobeniusNumber compute the Frobenius number of \(S\).
gap> FrobeniusNumber(S) + 1;
9\(\diamond\) Let \(S = \langle 4, 20, 21, 22 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 20, 21, 22);
<Numerical semigroup with 4 generators>The function SmallElements returns a list with the left elements and the conductor of \(S\).
gap> SmallElements(S);
[ 0, 4, 8, 12, 16, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40 ]We obtain \(C(S) = 40\), we can check it with the function Conductor.
gap> Conductor(S);
40References
https://gap-packages.github.io/ numericalsgps.