Saturated closure
Definition
Let \(S\) be a numerical semigroup and \(\mathcal{O}(S)\) the set of oversemigroups of \(S\). It is defined the saturated closure of \(S\) as the smallest saturated numerical semigroup containing \(S\), and it is denoted by \(Sat(S)\).
It can be proven that the intersection of saturated semigroups is also a saturated semigroup, from which it follows that the saturated closure is the intersection of all saturated numerical semigroups containing it, and since \(\mathcal{O}(S)\) is finite, the intersection is also finite.
Examples
\(\circ\) Let \(S = \langle 3, 7, 11 \rangle = \{0, 3, 6, 7, 9, \rightarrow\}\). Since \(7 + 7 - 2\cdot3 = 8 \not \in S\), we have that \(S\) is not saturated. The set of oversemigroups is
\[ \mathcal{O}(S) = \{\mathbb{N}, \{0, 2, \rightarrow \}, \{0, 3, \rightarrow \}, \{0, 3, 4, 6, \rightarrow\}, \{0, 3, 5, \rightarrow \}, \{0, 3, 6, \rightarrow \}, S\} \]
Moreover, the saturated semigroups are
\[ T_0 = \mathbb{N}, T_1 = \{0, 2, \rightarrow \}, T_2 = \{0, 3, \rightarrow \}, T_3 = \{0, 3, 5, \rightarrow \}, T_4 = \{0, 3, 6, \rightarrow \}, \]
Therefore, the saturated closure of \(S\) is \(A = \bigcap_{i = 0}^4 T_i = \{0, 3, 6, \rightarrow \} = S \cup \{8\}\). In this case, the saturated closure of \(S\) is the numerical semigroup \(S\) to which its Frobenius number has been added.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 12, 25, 38, 40 \rangle\), in GAP:
gap> S := NumericalSemigroup(12, 25, 38, 40);
<Numerical semigroup with 4 generators>The functions SaturatedClosure and SaturatedNumericalSemigroupClosure compute the saturated closure of a numerical semigroup.
gap> A := SaturatedClosure(S);
<Numerical semigroup>
gap> SaturatedClosure(S) = SaturatedNumericalSemigroupClosure(S);
true
gap> SmallElements(A);
[ 0, 12, 24 ]The function SmallElements returns a list with the left elements and the conductor of the numerical semigroup. Then, the saturated closure of \(S\) is \(A = \{0, 12, 24, \rightarrow \}\).
References
https://gap-packages.github.io/ numericalsgps.