Genus of a relative ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a relative ideal of \(S\). It is defined the genus of \(E\) as \(g(E) = |(m(E) + \mathbb{N}) \setminus E|\), where \(m(E)\) denotes the multiplicity of \(E\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(E = \{-14\} + S = \{-14, -9, -7, -5, -4, -2, 0, \rightarrow\}\). Clearly, \(m(E) = -14\), so \(g(E) = |(-14 + \mathbb{N}) \setminus E| = |\{-13, -12, -11, -10, -8, -6, -3, -1\}| = 8\).
\(\circ\) The definition of genus of a relative ideal is a generalization of the definition of genus of a numerical semigroup. Indeed, let \(S\) be a numerical semigroup with genus \(g\). In particular, \(S\) is a relative ideal of itself, then \(g(S) = |(m(S) + \mathbb{N}) \setminus S|\). The multiplicity of \(S\) (as relative ideal) is \(m(S) = 0\), therefore \(g(S) = |\mathbb{N} \setminus S| = g\).
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to genus of a relative ideal. However, given a relative ideal \(I\), the following function returns the genus of \(I\).
gap> GenusOfRelativeIdeal := function(I)
> local m, T;
> if not IsIdealOfNumericalSemigroup(I) then
> Error("The argument must be a relative ideal");
> fi;
> m := Minimum(I);
> T := m + NumericalSemigroup(1);
> return Length(Difference(T, I));
> end;
function( I ) ... end\(\diamond\) Let \(S = \langle 33, 37, 41, 45, 91 \rangle\) and \(I = \{-14, -11, -3, 5, 12 \} + S\), in GAP:
gap> S := NumericalSemigroup(33, 37, 41, 45, 91);
<Numerical semigroup with 5 generators>
gap> I := [-14, -11, -3, 5, 12] + S;
<Ideal of numerical semigroup>Applying the function defined above, the genus of the relative ideal \(I\) is
gap> GenusOfRelativeIdeal(I);
62From the definition, it can be deduced that \(g(I) = C(I) - m(I) + 1 - |N(I)|\), where \(C(I)\) is the conductor of \(I\), defined in the same way as the conductor of a numerical semigroup, and \(N(I)\) is the set of left elements of \(I\), again defined in the same way as the left elements of a numerical semigroup.
gap> Conductor(I) - Minimum(I) + 1 - Length(SmallElements(I));
62References
https://gap-packages.github.io/ numericalsgps.