Ratliff-Rush number of a proper ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a proper ideal of \(S\). It is defined the Ratliff-Rush number of \(I\), denoted by \(R(I)\), as the least integer \(n\) such that \(S \cap ( (n+1)I - nI)\) is the Ratliff-Rush closure of \(I\).
If \(r(I)\), that is, the minimum \(h \ge 1\) such that \((h + 1)I = m(I) + hI\), where \(m(I)\) is the multiplicity of \(I\), it can be proven that \(R(I) \le r(I)\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle\) and \(I = \{2, 4\} + S = \{2, 4, 5, 7, \rightarrow \}\). Clearly, \(m(I) = 2\), then we are looking for the smaller \(h \in \mathbb{N} \setminus \{0\}\) such that \((h+1)I = 2 + hI\). If we take \(h = 1\),
\[ 2I = I + I = \{4, 6, \rightarrow\} \ne \{4, 6, 7, 9, \rightarrow \} = 2 + I. \]
For \(h = 2\),
\[ 3I = (I + I) + I = \{6, 8, \rightarrow\} = 2 + 2I \]
Then, the reduction number of \(I\) is \(r(I) = 2\). Now, with few computations,
\[ S \cap (2I - I) = \{5, \rightarrow \} = S \cap (3I - 2I), \]
concluding that the Ratliff-Rush number of \(I\) is \(R(I) = 1\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 50, 54, 78, 115, 116, 119 \rangle\) and \(I = \{50, 78, 119\} + S\), in GAP:
gap> S := NumericalSemigroup(50, 54, 78, 115, 116, 119);
<Numerical semigroup with 6 generators>
gap> I := [50, 78, 119] + S;
<Ideal of numerical semigroup>Given a relative ideal \(I\) of a numerical semigroup, the functions RatliffRushNumber and RatliffRushNumberOfIdealOfNumericalSemigroup return the Ratliff-Rush number of \(I\).
gap> RatliffRushNumber(I);
6
gap> RatliffRushNumberOfIdealOfNumericalSemigroup(I) = RatliffRushNumber(I);
trueThen, the Ratliff-Rush closure of \(I\) is \(S \cap (7I - 6I)\).
\(\diamond\) Let \(S = \langle 39, 96, 105, 120, 130 \rangle\) and \(I = \{96, 169, 183, 195 \} + S\), in GAP:
gap> S := NumericalSemigroup(39, 96, 105, 120, 130);
<Numerical semigroup with 5 generators>
gap> I := [96, 169, 183, 195] + S;
<Ideal of numerical semigroup>Given a proper ideal \(I\), the functions AsymptoticRatliffRushNumber and AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup return the least integer \(n\) such that the Ratliff-Rush closure of \(mI\) equals to \(mI\) for all \(m \ge n\), that is, the asymptotic Ratliff-Rush number.
gap> AsymptoticRatliffRushNumber(I);
4
gap> RatliffRushNumber(I);
3It can be proven that the asymptotic Ratliff-Rush number is greater or equal to the Ratliff-Rush number.
References
https://gap-packages.github.io/ numericalsgps.