Pseudo-Frobenius number of a relative ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). If \(M = S \setminus \{0\}\) is the maximal ideal of \(S\), it is defined the set of pseudo-Frobenius numbers of \(E\), denoted by \(PF(E)\), as
\[ PF(E) = (E-M) \setminus E = \{x \in \mathbb{Z} ~ | ~ z \not \in E ~ \land ~ z + M \subseteq E\}, \]
where
\[ z + M = \{z + m ~ | ~ m \in M\}. \]
The cardinal of \(PF(E)\) is called the type of the relative ideal \(E\) and is a generalizations of the type of a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{ 0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(I = \{-1, -2\} + S\). Since \(I = (\{-1\} + S) \cup (\{-2\} + S)\), it is deduced that \(I = \{-2, -1, 3, \rightarrow\}\) and
\[ I - M = \{z \in \mathbb{Z} ~ | ~ z + M \subseteq I\} = \{-2, \rightarrow\}, \] concluding that \(PF(I) = \{0, 1, 2\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 19, 47, 50, 55, 68, 89 \rangle\) and \(I = \{3, 23, 37, 86\} + S\), in GAP:
gap> S := NumericalSemigroup(19, 47, 50, 55, 68, 89);
<Numerical semigroup with 6 generators>
gap> I := [3, 23, 37, 86] + S;
<Ideal of numerical semigroup>Given a relative ideal of a numerical semigroup \(I\), the functions PseudoFrobenius and PseudoFrobeniusOfIdealOfNumericalSemigroup return the set of pseudo-Frobenius numbers of \(I\).
gap> PseudoFrobenius(I);
[ 39, 65, 68, 81, 93, 101, 102, 104, 114 ]
gap> PseudoFrobeniusOfIdealOfNumericalSemigroup(I) = PseudoFrobenius(I);
trueReferences
https://gap-packages.github.io/ numericalsgps.