Dominant
Definition
Let \(S\) be a numerical semigroup with positive conductor \(C(S)\). It is defined the dominant of \(S\), denoted by \(d(S)\), as \(d(S) = \max \{s \in S ~ | ~ s < C(S)\}\).
Examples
\(\circ\) Let \(S = \langle 3, 7, 11 \rangle\). We have that \(9, 10, 11 \in S\) and \(8 \not \in S\), therefore the conductor is \(C(S) = 9\), and the dominant is \(d(S) = 7\). This numerical semigroup is also acute, since \(C(S) - d(S) = 2\).
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to dominant of a numerical semigroup. However, the following function in GAP returns the dominant of a given numerical semigroup.
gap> DominantOfNumericalSemigroup := function(S)
> local n;
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> if Conductor(S) <= 0 then
> Error("The conductor must be a positive integer");
> fi;
> n := Length(SmallElements(S));
> return SmallElements(S)[n-1];
>
> end;
function( S ) ... end\(\diamond\) Let \(S = \langle 7, 22, 47, 52 \rangle\), in GAP:
gap> S := NumericalSemigroup(7, 22, 47, 52);
<Numerical semigroup with 4 generators>If we compute the small elements of \(S\),
gap> SmallElements(S);
[ 0, 7, 14, 21, 22, 28, 29, 35, 36, 42, 43, 44, 47, 49, 50, 51,
52, 54, 56, 57, 58, 59, 61, 63, 64, 65, 66, 68 ]Then, the dominant of \(S\) is \(d(S) = 66\). With the function defined above,
gap> DominantOfNumericalSemigroup(S);
66References
https://gap-packages.github.io/ numericalsgps.