Blowup of a relative ideal
thatDefinition
Let \(S\) be a numerical semigroup and let \(E\) be a relative ideal of \(S\) with multiplicity ideal \(m(E)\) and reduction number \(r(E)\). It is defined the blowup of \(E\), also known as Lipman semigroup of \(S\), denoted by \(\mathcal{B}(E)\), as the relative ideal \(\mathcal{B}(E) = r(E)E - r(E)m(E)\), where \(nE = \{e_1 + e_2 + \cdots + e_n ~ | ~ e_1, \ldots, e_n \in E\}\) for all \(n \in \mathbb{N} \setminus \{0\}\) and \(A - B = \{z \in \mathbb{Z} ~ | ~ z + B \subseteq A\}\) for \(A,B \subseteq \mathbb{Z}\).
It can be proven that \(\mathcal{B}(E)\) is a numerical semigroup and the equality \(\mathcal{B}(E) = r(E)E - r(E)E\) holds.
Examples
\(\circ\) Let \(S = \langle 5, 9, 12, 13 \rangle\), \(I = \{14, 17\}\) and \(E = I + S = \{14, 17, 19, 22, 23, 24, 26, \rightarrow \}\). We have \(m(E) = 14\), and for \(h = 1\),
\[ E + E = \{28, 31, 33, 34, 36, \rightarrow\} \ne \{28, 31, 33, 36, 37, 38, 40, \rightarrow \} = 14 + E. \]
For \(h = 2\),
\[ 3E = (E + E) + E = \{42, 45, 47, 48, 50 , \rightarrow\} = 12 + 2E. \]
Therefore, \(r(E) = 2\) and the blowup of \(E\) is \(\mathcal{B}(E) = 2E - 2E = \{0, 3, 5, 6, 8, \rightarrow\}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 16, 21, 30, 47 \rangle\), \(I = \{37, 51, 102\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(16, 21, 30, 47);
<Numerical semigroup with 4 generators>
gap> I := [37, 51, 102];
[ 37, 51, 102 ]
gap> IS := I + S;
<Ideal of numerical semigroup>The functions BlowUp and BlowUpIdealOfNumericalSemigroup return the blowup of a given relative ideal.
gap> L := BlowUp(IS);
<Ideal of numerical semigroup>
gap> BlowUp(IS) = BlowUpIdealOfNumericalSemigroup(IS);
true
gap> Generators(L);
[ 0, 14, 28 ]Given a relative ideal \(L\), the function MinimalGenerators returns a list \(J\) such that \(L = J + S\). The function ReductionNumber returns the reduction number of a given relative ideal.
gap> r := ReductionNumber(IS);
2
gap> L = r*IS - r*IS;
true\(\diamond\) Given a numerical semigroup, the functions BlowUp and BlowUpOfNumericalSemigroup returns the blowup of the maximal ideal of \(S\), that is, \(M = S\setminus \{0\}\).
gap> S := NumericalSemigroup(33, 41, 46, 70, 83);
<Numerical semigroup with 5 generators>
gap> BM := BlowUp(S);
<Numerical semigroup with 5 generators>
gap> BlowUp(S) = BlowUpOfNumericalSemigroup(S);
true
gap> Generators(BM);
[ 8, 13, 33, 37, 50 ]Since the blowup of a relative ideal is a numerical semigroup, this process can be repeated. From this, process, it is defined the multiplicity sequence of \(S\).
References
https://gap-packages.github.io/ numericalsgps.