Dual of a relative ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). It is defined the dual of \(E\), denoted by \(E^*\), as
\[ E^* = S - E = \{z \in \mathbb{Z} ~ | ~ z + E \subseteq S\}. \]
It can be proven that \(E^*\) is also a relative ideal. This idea can be generalized considering \(J\) another relative ideal instead of \(S\).
Examples
\(\circ\) Let \(S = \langle 7, 9, 13, 17 \rangle = \{0, 7, 9, 13, 14, 16, 17, 18, 20, \rightarrow\}\) and \(I = \{-10, -7, 3\} + S\). If \(L = \{0, 7, 9, 13, 14, 16, 17, 18\}\) and \(L - n = \{l - n ~ | ~ l \in L\}\),
\[ I = (L - 10) \cup (L - 7) \cup (L + 3) \cup \{10, \rightarrow\} = \{-10, -7, -3, -1, 0, 2, 3, 4, 6, \rightarrow\}. \]
Then, taking into account that \(\{30, \rightarrow\} \subseteq S - I \subseteq \{10, \rightarrow\}\), it is obtained that
\[ I^* = S - I = \{23, 24, 27, 28, 30, \rightarrow\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 22, 30, 58, 78, 89, 93, 94, 103, 106 \rangle\) and \(I = \{3, 49, 56, 57, 60 \} + S\), in GAP:
gap> S := NumericalSemigroup(22, 30, 58, 78, 89, 93, 94, 103, 106);
<Numerical semigroup with 9 generators>
gap> I := [3, 49, 56, 57, 60] + S;
<Ideal of numerical semigroup>Given two relative ideals \(I,J\) of the same numerical semigroup \(S\); the function SubtractIdealsOfNumericalSemigroup returns \(J - I\). If we consider \(S\) as ideal, then it returns the dual of \(I\).
gap> I_2 := SubtractIdealsOfNumericalSemigroup(0+S, I);
<Ideal of numerical semigroup>
gap> L := MinimalGenerators(I_2);
[ 55, 63, 91, 103, 112, 117, 119, 120, 122, 127, 130, 131, 138, 146 ]The function MinimalGenerators returns the minimal set of generators. Then, \(I^* = L + S\).
References
https://gap-packages.github.io/ numericalsgps.