Proper ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a subset of \(\mathbb{Z}\). It is said that \(E\) is a proper ideal, an integral ideal, or simply an ideal of \(S\), if \(E\) is a relative ideal of \(S\) and \(E \subseteq S\).
Examples
\(\circ\) Let’s prove that \(E = S^*\) is an ideal of \(S\). Clearly \(S^* \subseteq S\) and \(S + S^* \subseteq S^*\). If we consider \(s = C(S)\), where \(C(S)\) denotes the conductor of \(S\), it is deduced that \(C(S) + S^* \subseteq S\) and \(S^*\) is an ideal. The ideal \(S^*\) is called the maximal ideal of \(S\).
\(\circ\) Let \(b \in \mathbb{N}\) an odd natural arbitrary but fixed, \(S = \langle 2, b \rangle = \{0, 2, 4, \ldots, b - 1, \rightarrow\}\) and \(E = \{0, b, \rightarrow \}\). Clearly, \(E\) is an ideal of \(S\). In general, given a numerical semigroup \(S\), the set \(E = \{C(S), \rightarrow\}\) is a proper ideal and is called the conductor ideal of \(S\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 10, 11, 15, 19 \rangle\), \(I = \{20, 21, 25\}\) and the relative ideal \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(10, 11, 15, 19);
<Numerical semigroup with 4 generators>
gap> I := [20, 21, 25];
[ 20, 21, 25 ]
gap> IS := I + S;
<Ideal of numerical semigroup>The functions IsIntegral and IsIntegralIdealOfNumericalSemigroup return true or false depending on whether an ideal of a semigroup is contained in the semigroup or not.
gap> IsIntegral(IS);
true
gap> IsIntegral(IS) = IsIntegralIdealOfNumericalSemigroup(IS);
true\(\diamond\) Given \(IS, JS\) ideals of a numerical semigroup \(S\) with \(JS \subseteq IS\), the functions Difference and DifferenceOfIdealsOfNumericalSemigroup return the set \(I \setminus J\).
gap> IsSubset(IS, JS);
true
gap> IS := [-4, 7, 22, 35, 64] + S;
<Ideal of numerical semigroup>
gap> JS := [-4, 21, 28, 43, 50, 51] + S;
<Ideal of numerical semigroup>
gap> Difference(IS, JS);
[ 7, 14, 22, 29, 36, 44 ]Given a numerical semigroup \(S\) and a subset \(X\) of \(S\), the function IsComplementOfIntegralIdeal returns true or false depending on whether \(X\) is a complement of a proper ideal or not.
gap> IsComplementOfIntegralIdeal([0, 36, 46, 50, 51],S);
false
gap> IsComplementOfIntegralIdeal([0, 46, 83],S);
trueReferences
https://gap-packages.github.io/ numericalsgps.