Embedding dimension
Definition
Let \(S\) be a numerical semigroup. It is well known that every numerical semigroup admits a unique finite minimal system of generators \(P(S) = \{n_1, n_2, \ldots, n_e\}\) for some \(e \in \mathbb{N} \setminus \{0\}\). The embedding dimension is defined as the cardinal of the minimal system of generators, and it is denoted by \(e(S), |P(S)|\) or \(d(S)\).
It can be proven that \(e(S) \le m(S)\), where \(m(S)\) denotes the multiplicity of \(S\). If \(e(S) = m(S)\), it is said that \(S\) is a numerical semigroup with maximal embedding dimension.
Examples
\(\circ\) Let \(S = \{0, 3, 6, 7, 9, 10, 12, \rightarrow\}\). First, we will compute the minimal system of generators of \(S\). The multiplicity of \(S\) is \(m(S) = 3\) and is always a minimal generator. On the other hand, the number \(7\) cannot be generated by \(3\), then \(7\) is another minimal generator. Considering the generators system \(B = \{3, 7\}\), we obtain \(S\). Therefore, \(e(S) = |B| = 2\).
\(\circ\) Let us consider \(m \in \mathbb{N} \setminus \{0\}\) arbitrary but fixed and \(S = \langle m, m+1, m+2, \ldots, 2m - 1 \rangle\). It is easy to see that \(B = \{m, m+1, m+2, \ldots, 2m - 1\}\) is a minimal system of generators of \(S\). Therefore \(e(S) = m = m(S)\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 5, 12, 15, 17, 18, 19, 29, 35, 50 \rangle\), in GAP:
gap> S := NumericalSemigroup(5, 12, 15, 17, 18, 19, 29, 35, 50);
<Numerical semigroup with 9 generators>The functions EmbeddingDimension and EmbeddingDimensionOfNumericalSemigroup compute the embedding dimension of \(S\).
gap> EmbeddingDimension(S);
4
gap> EmbeddingDimensionOfNumericalSemigroup(S);
4Using the MinimalGenerators function, we can also obtain the minimal system of generators of \(S\).
gap> MinimalGenerators(S);
[ 5, 12, 18, 19 ]References
https://gap-packages.github.io/ numericalsgps.