Maximal ideal
Definition
Let \(S\) be a numerical semigroup. It is defined the maximal ideal of \(S\) as the subset \(M = S^*\), and it is denoted by \(M(S)\). This definition is motivated by the definitions of relative ideal of \(S\) and proper ideal of \(S\).
Examples
\(\circ\) Let \(S\) a numerical semigroup and \(E\) a proper ideal of \(S\). By definition, \(E \subseteq S\), and if \(0 \in E\), then \(S \subseteq S + E\). On the other hand, by definition of proper ideal, \(S + E \subseteq E\), concluding that \(E = S\). In conclusion, every proper ideal of \(S\) other than \(S\) is a subset of \(S^*\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 15, 20 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 15, 20);
<Numerical semigroup with 3 generators>There are two ways to compute the maximal ideal of a numerical semigroup, these are with the functions MaximalIdeal and MaximalIdealOfNumericalSemigroup.
gap> I := MaximalIdeal(S);
<Ideal of numerical semigroup>
gap> J := MaximalIdealOfNumericalSemigroup(S);
<Ideal of numerical semigroup>
gap> I = J;
trueReferences
https://gap-packages.github.io/ numericalsgps.