Translation of a relative ideal
Definition
Let \(S\) be a numerical semigroup and let \(I,J\) be relative ideals of \(S\), with \(J\) minimally generated by \(\{j_1, \ldots, j_n\}\). It is said that \(I\) is a translation of \(J\) if there exists \(k \in \mathbb{Z}\) such that \(I\) is generated by \(\{j_1 + k, \ldots, j_n + k\}\). In this case, it is said that \(I\) and \(J\) are equivalents.
Examples
\(\circ\) If \(J\) is a relative ideal of a numerical semigroup \(I\) minimally generated by \(\{j_1, \ldots, j_n\}\), then for any \(k \in \mathbb{Z}\), the translation of \(J\) by \(k\) is minimally generated by \(\{j_1 + k, \ldots, j_n + k\}\). Indeed, if \(I\) is a translation of \(J\) by \(k\), Then its set of minimal generators is \(I \setminus (I + M)\) and \(j + k \in I \setminus (M + I)\) if, and only if, \(j \in J \setminus (M + J)\), concluding that \(j_i + k\) is minimal and \(\{j_1+k, \ldots, j_n + k\}\) is the set of minimal generators of \(I\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 25, 26, 33, 65, 80, 81 \rangle\), in GAP:
gap> S := NumericalSemigroup(25, 26, 33, 65, 80, 81);
<Numerical semigroup with 6 generators>
gap> I := [8, 17, 38, 45, 65] + S;
<Ideal of numerical semigroup>Given a relative ideal of a numerical semigroup and an integer \(k\), the function TranslationOfIdealOfNumericalSemigroup returns the translation of \(I\) by \(k\).
gap> J := TranslationOfIdealOfNumericalSemigroup(15, I);
<Ideal of numerical semigroup>The translation can be also computed with the \(+\) operator.
gap> J = I + 15;
trueReferences
https://gap-packages.github.io/ numericalsgps.