Reduction number
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a proper ideal of \(S\) with multiplicity \(m(E)\). It is defined the reduction number of \(E\), denoted by \(r(E)\), as the value
\[ r(E) = \min \{h \ge 1 ~ | ~ (h+1)E = m(E) + hE \}, \]
where \(nE = \{e_1 + e_2 + \cdots + e_n ~ | ~ e_1, \ldots, e_n \in E\}\) for all \(n \in \mathbb{N} \setminus \{0\}\). It can be proven that \(r(E)\) exists for any proper ideal \(E\). If \(r(E) = 1\), that is, if \(2E = m(E) + E\), it is said that \(E\) is stable.
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle\), \(I = \{2, 4\}\) and \(E = I + S = \{2, 4, 5, 7, \rightarrow \}\). Clearly, \(m(E) = 2\), then we are looking for the smaller \(h \in \mathbb{N} \setminus \{0\}\) such that \((h+1)E = 2 + hE\). If we take \(h = 1\),
\[ 2E = E + E = \{4, 6, \rightarrow\} \ne \{4, 6, 7, 9, \rightarrow \} = 2 + E. \]
For \(h = 2\),
\[ 3E = (E + E) + E = \{6, 8, \rightarrow\} = 2 + 2E \]
To sum up, the reduction number of \(E\) is \(r(E) = 2\) and is not stable.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 11, 14, 20 \rangle\), \(I = \{22, 45, 53\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(11, 14, 20);
<Numerical semigroup with 3 generators>
gap> I := [22, 45, 53];
[ 22, 45, 53 ]
gap> IS := I + S;
<Ideal of numerical semigroup>The function ReductionNumber returns the reduction number of a proper ideal.
gap> ReductionNumber(IS);
2Then, \(3IS = m(IS) + 2IS\), let us check it. The function Minimum returns the multiplicity of a relative ideal.
gap> Minimum(IS);
22
gap> 3*IS = 22 + 2*IS;
trueIf we consider the case \(h = 1\),
gap> 2*IS = 22 + IS;
falseReferences
https://gap-packages.github.io/ numericalsgps.