Multiplicity
Definition
Let \(S\) be a numerical semigroup. It is said that \(s \in S\) is the multiplicity of \(S\) if it is the smallest non-zero element of \(S\), and it is denoted by \(m(S)\) or \(\mu(S)\). Equivalently, the multiplicity of \(S\) is defined as the minimum of the minimal generator system of S.
It can be proven that the multiplicity is an upper bound of the embedding dimension of \(S\), it is always a minimal generator and does not determine the numerical semigroup.
Examples
\(\circ\) Let us consider a non-zero \(m \in \mathbb{N}\) arbitrary but fixed and \(S = \{0, m, \rightarrow\}\). Obviously \(m(S) = m\). This example shows that for any \(n \in \mathbb{N} \setminus \{0\}\), there exists a numerical semigroup \(S\) so that \(m(S) = n\).
\(\circ\) Let \(S_1 = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(S_2 = \{0, 5, \rightarrow\}\), obviously \(m(S_1) = 5 = m(S_2)\) and \(S_1 \ne S_2\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 3, 7, 9, 13 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 7, 9, 13);
<Numerical semigroup with 4 generators>The functions Multiplicity and MultiplicityOfNumericalSemigroup compute the multiplicity of a numerical semigroup.
gap> Multiplicity(S);
3
gap> MultiplicityOfNumericalSemigroup(S);
3We can check if \(m(S) = 3\) is a minimal generator with the function MinimalGenerators, which returns the set of minimal generators of the numerical semigroup.
gap> MinimalGenerators(S);
[3, 7]References
https://gap-packages.github.io/ numericalsgps.