Elasticity of an element
Definition
Let \(S\) be a numerical semigroup and \(s \in S\). If \(\mathbf{L}(s)\) denotes the set of lengths of \(s\), it is defined the elasticity of \(s\), denoted by \(\rho(s)\), as
\[ \rho(s) = \frac{\max \mathbf{L}(s)}{\min \mathbf{L}(s)}. \]
From the elasticity of an element it is defined the elasticity of a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 5, 9, 16 \rangle\), \(s = 56 \in S\) and \(\mathbf{Z}(s)\) the set of factorizations of \(s\) in \(S\). If \((\alpha, \beta, \gamma) \in \mathbf{Z}(s)\), then \(0 \le \alpha \le 11, 0 \le \beta \le 6, 0 \le \gamma \le 3\), from which it follows that \(\mathbf{Z}(56) = \{(4,4,0), (8,0,1), (3,1,2)\}\). Its set of lengths is \(\mathbf{L}(56) = \{6, 8, 9\}\) and the elasticity of \(s = 56\) is \(\rho(56) = 3/2\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 14, 31, 68, 81 \rangle\), in GAP:
gap> S := NumericalSemigroup(14, 31, 68, 81);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n \in S\), the functions Elasticity and ElasticityOfFactorizationsElementWRTNumericalSemigroup return the elasticity of \(n\) in \(S\). Given only a numerical semigroup \(S\), this function returns the elasticity of \(S\).
gap> Elasticity(250, S);
2
gap> Elasticity(S, 250);
2
gap> Elasticity(250, S) = ElasticityOfFactorizationsElementWRTNumericalSemigroup(250, S);
true
gap> Elasticity(S);
81/14The function LengthsOfFactorizationsElementWRTNumericalSemigroup returns the set of lengths of an element of a numerical semigroup.
gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(250, S);
[ 7, 8, 13, 14 ]References
https://gap-packages.github.io/ numericalsgps.