Maximal embedding dimension system of generators
Definition
Let \(S\) be a maximal embedding dimension numerical semigroup and \(M = \{m_1, \ldots, m_k\}\) a set of positive integers with \(gcd(M) = 1\) and \(m_1 < \ldots < m_k\). It is said that \(M\) is a maximal embedding dimension system of generators of \(S\) (or a MED system of generators of \(S\)) if the MED closure of \(M\) is \(S\).
It can be proven that for any MED numerical semigroup \(S\), there exists a finite and unique minimal MED system of generators \(M\) such that the MED closure of \(M\) is \(S\). Moreover, \(M\) is a subset of the minimal set of generators of \(S\). This definition is a particular case of V-monoids generated by a subset.
Examples
\(\circ\) Let \(S = \langle 4, 6, 7, 9 \rangle\). Since \(e(S) = m(S) = 4\), where \(e(S)\) is the embedding dimension and \(m(S)\) is the multiplicity of \(S\) respectively, we have that \(S\) has maximal embedding dimension. From the condition \(gcd(M) = 1\), the candidates to be the MED system of generators of \(S\) are
\[ M_1 = \{4, 7\}, ~~ M_2 = \{4, 9\}, ~~ M_3 = \{6, 7\}, ~~ M_4 = \{7,9\}, \]
\[ M_5 = \{4, 6, 7\}, ~~ M_6 = \{4, 6, 9\}, ~~ M_7 = \{6, 7, 9\}, ~~ M_8 = \{4, 6, 7, 9\}. \]
If \(S\) is the MED closure of \(M_j\) with \(j \in \{1, \ldots, 8\}\), then \(\min M_j = m(S) = 4\), obtaining that \(M_1, M_2, M_5, M_6, M_8\) are the only possibilities. From this, it is deduced that the MED system of generators is \(M_5 = \{4,6,7\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 6, 9, 10, 11, 13, 14 \rangle\), in GAP:
gap> S := NumericalSemigroup(6, 9, 10, 11, 13, 14);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\), the function IsMED returns true if \(S\) is a MED numerical semigroup and false otherwise.
gap> IsMED(S);
trueMoreover, the function MinimalMEDGeneratingSystemOfMEDNumericalSemigroup returns the MED system of generators of \(S\).
gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup(S);
[ 6, 9, 10, 11 ]References
https://gap-packages.github.io/ numericalsgps.