Type
Definition
Let \(S\) be a numerical semigroup and let \(PF(S)\) be the set of pseudo-Frobenius numbers of \(S\). The type of \(S\) is defined as the cardinal of \(PF(S)\).
Examples
\(\circ\) Let \(S = \langle 4, 6, 9, 13 \rangle = \{0, 4, 6, 8, 9, 10, 12, \rightarrow \}\), let us compute the pseudo-Frobenius numbers of \(S\). The positive integers that do not belong to \(S\) are \(\mathbb{N} \setminus S = \{1,2,3,5,7,11\}\) and as \(4 \in S\), the candidates for pseudo-Frobenius numbers belongs to the set \(\{-4, -3, -2, -1, 1,2,3,5,7,11 \}\). Checking the condition of pseudo-Frobenius number on every candidate, it is conclude that \(PF(S) = \{11\}\) and \(t(S) = 1\).
\(\circ\) The type of a numerical semigroup is \(1\) if, and only if, \(2g(S) = C(S)\), where \(g(S)\) is the genus of \(S\) and \(C(S)\) the conductor of \(S\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 7, 9, 10, 15, 20 \rangle\), in GAP:
gap> S := NumericalSemigroup(7, 9, 10, 15, 20);
<Numerical semigroup with 5 generators>To compute the type of \(S\) we can apply the function Type or TypeOfNumericalSemigroup.
gap> Type(S);
4
gap> Type(S) = TypeOfNumericalSemigroup(S);
trueApplying the function PseudoFrobenius, we are able to obtain explicitly the pseudo-Frobenius numbers of \(S\).
gap> PseudoFrobenius(S);
[ 8, 11, 12, 13 ]References
https://gap-packages.github.io/ numericalsgps.