Elasticity of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is defined the elasticity of \(S\), denoted by \(\rho(S)\), as
\[ \rho(S) = \sup \{\rho(s) ~ | ~ s \in S\}, \]
where \(\rho(s)\) denotes the elasticity of s in \(S\). If \(S\) is minimally generated by \(\{n_1, n_2, \ldots, n_p\}\) with \(n_1 < n_2 < \cdots < n_p\), it can be proven that
\[ \rho(S) = \frac{n_p}{n_1}. \]
Examples
\(\circ\) Let \(S = \langle 17, 42, 89, 104 \rangle\). As \(B = \{17, 42, 89, 104\}\) is a minimal set of generators, the elasticity of \(S\) is \(\rho(S) = 104/17\). If we take \(s = 17 \cdot 104\), we have
\[ 17 \cdot 104 = 104 \cdot 17 + 0 \cdot 42 + 0 \cdot 89 + 0 \cdot 104 = 0 \cdot 17 + 0 \cdot 42 + 0 \cdot 89 + 17 \cdot 104. \]
Then, \((104,0,0,0)\) and \((0,0,0,17)\) are factorizations of \(17 \cdot 104\) in \(S\) with lengths \(104\) and \(17\), respectively. Therefore,
\[ \frac{104}{17} \le \rho(17 \cdot 104) \le \rho(S) = \frac{104}{17}. \]
In conclusion, \(\rho(17 \cdot 104) = 104/17\) and the elasticity of \(S\) is reached by \(s = 17 \cdot 104\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 18, 43, 92, 105 \rangle\), in GAP;
gap> S := NumericalSemigroup(18, 43, 92, 105);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the functions Elasticity and ElasticityOfNumericalSemigroup return the elasticity of \(S\). In the function Elasticity, if we also give an element \(n \in S\), the function returns the elasticity of \(n\) in \(S\).
gap> Elasticity(S);
35/6
gap> Elasticity(S) = ElasticityOfNumericalSemigroup(S);
true
gap> Elasticity(S, 415);
6/5
gap> Elasticity(S, 1890);
35/6References
https://gap-packages.github.io/ numericalsgps.