Type of a relative ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a relative ideal of \(S\). It is defined the type of \(E\) as \(t(E) = |(E - M) \setminus E|\), where \(M = S^*\) is the maximal ideal of \(S\) and
\[ E - M = \{z \in \mathbb{Z} ~ | ~ z + M \subseteq E\}. \]
Examples
\(\circ\) The definition of type of a relative ideal is a generalization of the definition of type of a numerical semigroup. Indeed, let \(S\) be a numerical semigroup with type \(t\). In particular, \(S\) is a relative ideal of itself, then \(t(S) = |(S - S^*) \setminus S|\). By definition,
\[ S - S^* = \{z \in \mathbb{Z} ~ | ~ z + S^* \subseteq S\}, \]
where it is deduced that \((S - S^*) \setminus S\) is the set of pseudo-Frobenius number of \(S\). In conclusion, \(t(S) = t\).
\(\circ\) Let \(S = \langle 5, 7, 11 \rangle = \{0, 5, 7, 10, 11, 12, 14, \rightarrow \}\) and \(E = \{-4\} + S = \{-4, 1, 3, 6, 7, 8, 10, \rightarrow\}\). We have,
\[ E - S^* = \{z \in \mathbb{Z} ~ | ~ z + S^* \subseteq E\} \]
The candidates are in the set \(\{-9, -8, \ldots, 9\} \setminus E\), and checking the condition on each element, we get \(E - S^* = \{5, 9\}\) and \(t(E) = 2\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 10, 11, 15, 19 \rangle\) and \(IS = \{20, 21, 25\} + S\), in GAP:
gap> S := NumericalSemigroup(10, 11, 15, 19);
<Numerical semigroup with 4 generators>
gap> I := [20, 21, 25];
[ 20, 21, 25 ]
gap> IS := I + S;
<Ideal of numerical semigroup>The function Type returns the type of a relative ideal of a numerical semigroup. This function also returns the type of a numerical semigroup.
gap> Type(S);
4
gap> Type(IS);
2If we want to know the elements of \((E - M) \setminus E\), we can apply the function PseudoFrobenius. This function also returns the set of pseudo-Frobenius numbers of a numerical semigroup.
gap> PseudoFrobenius(IS);
[ 34, 48 ]
gap> PseudoFrobenius(S);
[ 23, 24, 27, 28 ]References
https://gap-packages.github.io/ numericalsgps.