Frobenius number
Definition
Let \(S\) be a numerical semigroup. It is defined the Frobenius number of \(S\), denoted by \(F(S)\) or \(g(S)\), as the maximum of the set \(\mathbb{N} \setminus S\). Equivalently, the Frobenius number is the largest number that does not belong to \(S\) or the unique element in \(\mathbb{N} \setminus S\) so that for any \(n \in \mathbb{N} \setminus \{0\}\), we have that \(F(S) + n \in S\).
There is a obvious relation between the conductor and the Frobenius number, that is, \(F(S) + 1 = C(S)\). The Frobenius number does not determine in general the numerical semigroup, that means, it may exist more than one numerical semigroup with the same Frobenius number.
Examples
\(\circ\) Let us consider a non-zero \(m \in \mathbb{N}\) arbitrary but fixed and \(S = \{0, m, \rightarrow\}\). Obviously, \(F(S) = m-1\) and in this case every element of \(S \setminus \{0\}\) is greater than the Frobenius number. This kind of numerical semigroups are the only ones that satisfy this property. This example shows that for any \(n \in \mathbb{N} \setminus \{0\}\), there exists a numerical semigroup \(S\) so that \(F(S) = n\).
\(\circ\) 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 \(F(S_1) = 13 = F(S_2)\) and \(S_1 \ne S_2\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 3, 10, 16, 19 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 10, 16, 19);
<Numerical semigroup with 4 generators>The functions FrobeniusNumber and FrobeniusNumberOfNumericalSemigroup compute the Frobenius number of \(S\).
gap> FrobeniusNumber(S);
17Taking into account the equality \(F(S) + 1 = C(S)\), we can compute the Frobenius number from the conductor. The function Conductor computes the conductor of \(S\).
gap> Conductor(S) - 1;
17\(\diamond\) Let \(S = \langle 4, 20, 21, 22 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 20, 21, 22);Using the function SmallElements we can obtain 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 \(F(S) = C(S) - 1 = 40 - 1 = 39\).
gap> FrobeniusNumber(S);
39If we want to know how many numerical semigroups have Frobenius number \(f \in \mathbb{N}\), we have the function NumericalSemigroupsWithFrobeniusNumberFG that returns the set of these numerical semigroups. For \(f = 15\),
gap> Length(NumericalSemigroupsWithFrobeniusNumberFG(15));
200References
https://gap-packages.github.io/ numericalsgps.