Trace ideal
Definition
Let \(S\) be a numerical semigroup and let \(K\) be its canonical ideal. It is defined the trace ideal of \(S\), denoted by \(tr(S)\), as \(tr(S) = K + (S - K)\), where
\[ S - K = \{z \in \mathbb{Z} ~ | ~ z + K \subseteq S\}, \] and
\[ K + (S - K) = \{k + s' ~ | ~ k \in K, s' \in S-K \}. \]
Examples
\(\circ\) Let \(S = \langle 7, 9, 11, 12 \rangle = \{0, 7, 9, 11, 12, 14, 16, 18, \rightarrow\}\). By definition, its canonical ideal is
\[ K(S) := \{x \in \mathbb{Z} ~ | ~ F(S) - x \not \in S \}, \] where \(F(S)\) denotes the Frobenius number of \(S\). Since \(F(S) = 17\) and \(S \subseteq K(S) \subseteq \mathbb{N}\), it is deduced that
\[ K(S) = \{0, 2, 4, 7, 9, 11, 12, 13, 14, 15, 16, 18, \rightarrow\}. \] Moreover, since \(0 \in K(S)\), then \(S - K \subseteq S\) and taking into account that \(\{18, \rightarrow\} \subseteq K-S\), it is obtained the following:
\[ S - K = \{ 7, 12, 14, 16, 18, \rightarrow \}. \] Finally,
\[ \begin{align*} tr(S) &= K + (S - K) = \{0, 2, 4, 7, 9, 11, 12, 13, 14, 15, 16, 18, \rightarrow\} + \{ 7, 12, 14, 16, 18, \rightarrow \} \\ & = (\{0, 2, 4, 7, 9\} + \{ 7, 12, 14, 16\}) \cup \{18, \rightarrow\} = \{ 7, 9, 11, 12, 14, 16, 18, \rightarrow\}. \end{align*} \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 12, 15, 17, 20, 28, 38 \rangle\), in GAP:
gap> S := NumericalSemigroup(12, 15, 17, 20, 28, 38);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\), the functions TraceIdeal and TraceIdealOfNumericalSemigroup return the trace ideal of \(S\).
gap> tr := TraceIdeal(S);
<Ideal of numerical semigroup>
gap> TraceIdealOfNumericalSemigroup(S) = tr;
true
gap> SmallElements(tr);
[ 27, 28, 29, 30, 32, 34 ]The function SmallElements returns a list with the small elements of the given ideal. Then, \(tr(S) = \{27, 28, 29, 30, 32, 34, \rightarrow\}\).
References
https://gap-packages.github.io/ numericalsgps.