Ratio
Definition
Let \(S\) be a numerical semigroup other than \(\mathbb{N}\) and minimally generated by \(P = \{n_1, \ldots, n_p\}\) with \(n_1 < n_2 < \cdots < n_p\). It is defined the ratio of \(S\) as \(n_2\), and it is denoted by \(r(S)\).
Examples
\(\circ\) Let \(S = \langle 6, 12, 14, 15 \rangle\). Since \(12 = 2 \cdot 6\), the set \(B = \{6, 12, 14, 15\}\) is not a minimal system of generators and \(12\) is not the ratio. It is easy to deduce that \(B\setminus \{12\}\) is minimal. Then, the ratio of \(S\) is \(r(S) = 14\).
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to ratio of a numerical semigroup. However, given a numerical semigroup, the following function returns its ratio.
gap> RatioNumericalSemigroup := function(S)
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> if 1 in S then
> Error("The Numerical Semigroup must be other than N");
> fi;
> return MinimalGenerators(S)[2];
> end;
function( S ) ... end\(\diamond\) Let \(S = \langle 18, 36, 37, 40, 72 \rangle\), in GAP:
gap> S := NumericalSemigroup(18, 36, 37, 40, 72);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the function MinimalGenerators returns a list with the minimal generators of \(S\).
gap> MinimalGenerators(S);
[ 18, 37, 40 ]Then, the ratio of \(S\) is \(r(S) = 37\). With the function defined above,
gap> RatioNumericalSemigroup(S);
37References
https://gap-packages.github.io/ numericalsgps.