Threshold interval
Definition
Let \(S\) be a numerical semigroup, \(m(S)\) the multiplicity of \(S\) and \(C(S)\) the conductor of \(S\). It is defined the threshold interval of \(S\) as \([C(S), C(S) + m(S)) = \{C(S), C(S) + 1, \ldots, C(S) + m(S) - 1\}\).
Examples
\(\circ\) Let \(S = \langle 6, 8, 9 \rangle\). With some calculations, it is deduced that \(S = \{0, 6, 8, 9, 12, 14, 15, 16, 17, 18, 20, \rightarrow \}\), then \(m(S) = 6\) and \(C(S) = 20\). The threshold interval of \(S\) is \(\{20, 21, \ldots, 25\}\).
Examples with GAP
Nowadays, there are no functions in NumericalSgps related to threshold interval of a numerical semigroup. However, given a numerical semigroup \(S\), the following function returns the threshold interval of \(S\).
gap> ThresholdIntervalOfNumericalSemigroup := function(S)
> local c, m;
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> c := Conductor(S);
> m := Multiplicity(S);
> return [c..(c + m - 1)];
> end;
function( S ) ... end\(\diamond\) Let \(S = \langle 11, 17, 42, 71 \rangle\), in GAP:
gap> S := NumericalSemigroup(11, 17, 42, 71);
<Numerical semigroup with 4 generators>From the function defined above,
gap> ThresholdIntervalOfNumericalSemigroup(S);
[ 92 .. 102 ]Therefore, the threshold interval of \(S\) is \(\{92, 93, \ldots, 102\}\).
References
https://gap-packages.github.io/ numericalsgps.