Maximal embedding dimension closure
Definition
Let \(S\) be a numerical semigroup and let \(\mathcal{O}(S)\) be the set of oversemigroups of \(S\). It is defined the maximal embedding dimension closure (or MED closure) of \(S\), denoted by \(MED(S)\), as the smallest MED numerical semigroup containing \(S\).
Examples
\(\circ\) Let \(S = \langle 4, 7, 10 \rangle\). The maximal embedding dimension oversemigroups of \(S\) are
\[ S_1 = \mathbb{N}, ~ S_2 = \langle 2,3 \rangle, ~ S_3 = \langle 2, 5 \rangle, ~ S_4 = \langle 2, 7 \rangle, ~ S_5 = \langle 3, 4, 5 \rangle, \]
\[ S_6 = \langle 4, 5, 6, 7 \rangle, ~ S_7 = \langle 4, 6, 7, 9 \rangle, ~ S_8 = \langle 4,7,9,10 \rangle, ~ S_9 = \langle 4, 7, 10, 13 \rangle. \]
Therefore, the MED closure of \(S\) is
\[ MED(S) = \bigcap_{i = 1}^9 S_i = \langle 4, 7, 10, 13 \rangle = \{0, 4, 7, 8, 10, \rightarrow \}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 8, 10, 12, 15\rangle\), in GAP:
gap> S := NumericalSemigroup(8, 10, 12, 15);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the function MEDClosure returns the MED closure of \(S\).
gap> T := MEDClosure(S);
<Numerical semigroup>
gap> MinimalGenerators(T);
[ 8, 10, 12, 14, 15, 17, 19, 21 ]The function MinimalGenerators returns the minimal set of generators of the given numerical semigroup. Therefore,
\[ MED(S) = \langle 8, 10, 12, 14, 15, 17, 19, 21 \rangle. \]
References
https://gap-packages.github.io/ numericalsgps.