Reduction of a proper ideal
Definition
Let \(S\) be a numerical semigroup and \(H,E\) two proper ideals of \(S\) with \(H \subseteq E\). It is said that \(H\) is a reduction of \(E\) if, for some \(h \in \mathbb{N} \setminus \{0\}\), the equality \((h+1)E = H + hE\) holds, 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 \(H\) is a reduction of \(E\) if, and only if, \(m(E) + S \subseteq H \subseteq E\), where \(m(E)\) denotes the multiplicity of \(E\).
Examples
\(\circ\) Let \(S\) a numerical semigroup and \(E\) a proper ideal of \(S\) with multiplicity \(m(E)\). If \(A - B = \{z \in \mathbb{Z} ~ | ~ z + B \subseteq A\}\) for any \(A,B \subseteq \mathbb{Z}\), the following chain of relative ideals,
\[ S \subseteq E - m(E) \subseteq 2E - 2m(E) \subseteq \cdots \subseteq jE - jm(E) \subseteq \cdots \subseteq \mathbb{N}, \]
eventually stabilizes, since \(|\mathbb{N} \setminus S|\) is finite. Then, there exists \(h \in \mathbb{N} \setminus \{0\}\) such that \(hE - hm(E) = (h+1)E - (h+1)m(E)\), or equivalently, \((h+1)E = hE + m(E) = [m(E) + S] + hE\). In conclusion, \(m(E) + S\) is a reduction of \(E\).
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to reduction of proper ideals. However, given a numerical semigroup \(S\) and two proper ideals \(J,I\) of \(S\), the following function returns true if \(J\) is a reduction of \(I\) and false otherwise.
gap> IsReduction := function(S,J,I)
> if not IsNumericalSemigroup(S) then
> Error("First argument must be a Numerical Semigroup");
> fi;
> if not IsIdealOfNumericalSemigroup(J) then
> Error("Second argument must be a relative ideal of the first argument");
> fi;
> if not AmbientNumericalSemigroupOfIdeal(J) = S then
> Error("Second argument must be a relative ideal of the first argument");
> fi;
> if not IsIdealOfNumericalSemigroup(I) then
> Error("Third argument must be a relative ideal of the first argument");
> fi;
> if not AmbientNumericalSemigroupOfIdeal(I) = S then
> Error("Third argument must be a relative ideal of the first argument");
> fi;
> if not (IsIntegral(J) and IsIntegral(I)) then
> return false;
> fi;
> if IsSubset(J, Minimum(I) + S) and IsSubset(I,J) then
> return true;
> fi;
> return false;
> end;
function( S, J, I ) ... end\(\diamond\) Let \(S = \langle 13, 15, 22, 27 \rangle\), \(I = \{26, 28, 40, 41\} + S\) and \(J = \{26, 41\} + S\), in GAP:
gap> S := NumericalSemigroup(13, 15, 22, 27);
<Numerical semigroup with 4 generators>
gap> I := [26, 28, 40, 41] + S;
<Ideal of numerical semigroup>
gap> J := [26, 41] + S;
<Ideal of numerical semigroup>Since \(I,J \subseteq S\) are proper ideals and \(m(I) + S \subseteq J \subseteq I\), then \(J\) is a reduction of \(I\).
gap> IsReduction(S,J,I);
trueReferences
https://gap-packages.github.io/ numericalsgps.