Numerical semigroup with maximal embedding dimension
Definition
Let \(S\) be a numerical semigroup, \(e(S)\) the embedding dimension and \(m(S)\) the multiplicity of \(S\). Then, it is said that \(S\) has maximal embedding dimension (or MED) if \(e(S) = m(S)\).
It is well known that for every numerical semigroup, it holds the inequality \(e(S) \le m(S)\).
Examples
\(\circ\) Let \(S = \langle 3, 8 \rangle\). Clearly, \(m(S) = 3\) and \(e(S) = 2\), thus \(S\) does not have maximal embedding dimension.
\(\circ\) Let us consider \(a,b \in \mathbb{N}\) with \(gcd(a,b) = 1\) and \(S = \langle a, a+1, a+2, \ldots, a+b \rangle\). By construction of \(S\), \(m(S) = a\). To compute \(e(S)\), we divide into cases:
If \(b \ge a-1\), then \(B = \{a, a+1, \ldots, 2a - 1\}\) is the minimal set of generators of \(S\) and \(e(S) = a\). Therefore \(S\) has maximal embedding dimension.
If \(b < a-1\), then \(B = \{a, a+1, \ldots, a+b\}\) is the minimal set of generators of \(S\), \(e(S) = b+1\) and \(S\) does not have maximal embedding dimension.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 15, 20, 21, 22, 30, 31, 38 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 15, 20, 21, 22, 30, 31, 38);
<Numerical semigroup with 8 generators>Given a numerical semigroup \(S\), the functions IsMED and IsMEDNumericalSemigroup return true if \(S\) is MED (maximal embedding dimension) and false otherwise.
gap> IsMED(S);
true
gap> IsMEDNumericalSemigroup(S);
trueTherefore, \(e(S) = m(S)\). Given a numerical semigroup \(S\), the functions EmbeddingDimension and Multiplicity computes the embedding dimension and the multiplicity of \(S\).
gap> Multiplicity(S);
4
gap> EmbeddingDimension(S);
4References
https://gap-packages.github.io/ numericalsgps.