Pseudo-Frobenius number
Definition
Let \(S\) be a numerical semigroup. It is said that \(x \in \mathbb{Z}\) is a pseudo-Frobenius number if \(x \not \in S\) and \(x + s \in S\) for all \(s \in S^*\). The set of pseudo-Frobenius numbers of \(S\) is denoted by \(PF(S)\) or \(S'\).
\(PF(S)\) is always non-empty since \(F(S) \in PF(S)\), where \(F(S)\) is the Frobenius number of \(S\). The cardinal of \(PF(S)\) is the type of \(S\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 10 \rangle = \{0, 3, 5, 6, 8, \rightarrow \}\). The positive integers that do not belong to \(S\) are \(\mathbb{N} \setminus S = \{1, 2, 4, 7\}\), and as \(3 \in S\), the candidates for pseudo-Frobenius numbers belongs to the set \(\{-3, -2, -1, 1, 2, 4, 7 \}\). Checking the condition on each element, it is concluded that \(PF(S) = \{7\}\).
\(\circ\) The Frobenius number is the only pseudo-Frobenius number (the type 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 examples are 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>In order to compute the pseudo-Frobenius numbers of \(S\), we can apply the function PseudoFrobenius or PseudoFrobeniusOfNumericalSemigroup, which return the set of pseudo-Frobenius numbers of the numerical semigroup.
gap> PseudoFrobenius(S);
[ 8, 11, 12, 13 ]
gap> PseudoFrobenius(S) = PseudoFrobeniusOfNumericalSemigroup(S);
trueOn the other hand, the function FrobeniusNumber returns the Frobenius number of the numerical semigroup.
gap> FrobeniusNumber(S);
13It holds that for every numerical semigroup, \(F(S) = \max PF(S)\).
\(\diamond\) Given a list \(L\), the function NumericalSemigroupsWithPseudoFrobeniusNumbers returns all the numerical semigroups \(S\) such that \(PF(S) = L\). If we are looking for an example of a numerical semigroup \(S\) so that \(PF(S) = L\), we can apply ANumericalSemigroupsWithPseudoFrobeniusNumbers.
gap> pf := [ 58, 64, 75 ];
[ 58, 64, 75 ]
gap> Length(NumericalSemigroupsWithPseudoFrobeniusNumbers(pf));
561
gap> pf := [58, 64, 75];
[ 58, 64, 75 ]
gap> S := ANumericalSemigroupWithPseudoFrobeniusNumbers(pf);
<Numerical semigroup>
gap> MinimalGenerators(S);
[ 12, 30, 31, 35, 37, 41, 50, 56, 57 ]The function MinimalGenerators returns the set of minimal generators of a given numerical semigroup. Then, there is \(561\) numerical semigroups with set of pseudo-Frobenius numbers \(L = \{58, 64, 75\}\) and one of them is \(S = \langle 12, 30, 31, 35, 37, 41, 50, 56, 57 \rangle\).
References
https://gap-packages.github.io/ numericalsgps.