Irreducible Z-components
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). The definition of irreducible \(\mathbb{Z}-\)components of a relative ideal comes from the following result.
It is defined the irreducible \(\mathbb{Z}-\)components as the ideals \(-x_i + K\) that generates the factorization.
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle\), \(I = \{4, 5\}\) and \(E = I + S\). Making simple calculations, \(F(S) = 4\) and \(K(S) = \{0, 2, 3, 5, \rightarrow \}\), where \(K(S)\) denotes the standard canonical ideal of \(S\). We have that \(E = (-2 + K(S)) \cap (2 + K(S))\). Indeed,
\[ (-2 + K(S)) \cap (2 + K(S)) = \{-2, 0, 1, 3, \rightarrow \} \cap \{2, 4, 5, 7, \rightarrow\} \]
\[ = \{4, 5, 7, \rightarrow \} = \{4, 5\} + S = E. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 17, 21, 30, 45, 61 \rangle\), \(I = \{6, 10\}\) and \(F = I + S\), in GAP:
gap> S := NumericalSemigroup(17, 21, 30, 45, 61);
<Numerical semigroup with 5 generators>
gap> I := [6, 10];
[ 6, 10 ]
gap> F := I + S;
<Ideal of numerical semigroup>In order to compute the irreducible \(\mathbb{Z}-\)components of the relative ideal \(F\), the function IrreducibleZComponents returns a list with the ideals that generate the decomposition of the relative ideal.
gap> Irred := IrreducibleZComponents(F);
[ <Ideal of numerical semigroup>, <Ideal of numerical
semigroup>, <Ideal of numerical semigroup>,
<Ideal of numerical semigroup>, <Ideal of numerical semigroup> ]
gap> List(Irred, MinimalGenerators);
[ [ -8, -6, 6, 7, 10 ], [ -9, -7, 5, 6, 9 ], [ -11, -9, 3, 4, 7 ], [ -13, -11, 1, 2, 5 ], [ -24, -22, -10, -9, -6 ] ]References
https://gap-packages.github.io/ numericalsgps.