Z-irreducible ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a relative ideal of \(S\). It is said that \(E\) is \(\mathbb{Z}-\)irreducible if it cannot be expressed as a finite intersection of other relative ideals properly containing it.
It can be proven that every relative ideal has a unique finite factorization in \(\mathbb{Z}-\)irreducible components, which are called as the Z-irreducible components of \(E\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0,3,5, \rightarrow\}\), \(I = \{4, 5\}\) 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,
\[ (-2 + K(S)) \cap (2 + K(S)) = \{-2, 0, 1, 3, \rightarrow \} \cap \{2, 4, 5, 7, \rightarrow\} \]
\[ = \{4, 5, 7, \rightarrow \} = \{4, 5\} + S = E. \]
\(\circ\) Let \(S\) a numerical semigroup with Frobenius number \(F(S)\), \(K(S)\) the standard canonical ideal and \(E\) a canonical ideal of \(S\) with Frobenius number ideal \(F(E)\), let us prove that \(E\) is \(\mathbb{Z}-\)irreducible. If \(E\) is a canonical ideal, by definition \(\tilde{E} = K(S)\), where \(\tilde{E} = E + F(S) - F(E)\). Let us suppose that there exist \(K_1, K_2, \ldots, K_h\) relative ideals such that \(E \subseteq K_i\) for all \(i \in \{1, 2, \ldots, h\}\) and \(E = K_1 \cap K_2 \cap \ldots \cap K_h\). Then, there exists \(j \in \{1, 2, \ldots, h\}\) so that \(F(E) \not \in K_j\) and \(\{F(E) + 1, \rightarrow \} \subseteq K_j\), therefore \(F(K_j) = F(E)\). If \(E \subseteq K_j\), then
\[ K(S) = \tilde{E} = E + F(S) - F(E) \subseteq K_j + F(S) - F(E) = K_j + F(S) - F(K_j) = \tilde{K}_j, \]
but for any relative ideal, \(\tilde{K_j} \subseteq K(S)\). Hence, we have \(\tilde{E} = \tilde{K_j}\), which means that \(E = K_j\) and \(E\) is \(\mathbb{Z}-\)irreducible.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 14, 23, 30 \rangle\), \(I = \{6, 10\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(14, 23, 30);
<Numerical semigroup with 3 generators>
gap> I := [6, 10];
[ 6, 10 ]
gap> IS := I + S;
<Ideal of numerical semigroup>In order to compute the \(\mathbb{Z}-\)components of \(K\), the function IrreducibleZComponents returns a list with the relative ideals that generate the decomposition of a given relative ideal.
gap> Irred := IrreducibleZComponents(IS);
[ <Ideal of numerical semigroup>, <Ideal of numerical semigroup>, <Ideal of numerical semigroup> ]
gap> List(Irred, MinimalGenerators);
[ [ -36, -20 ], [ -52, -36 ], [ -102, -86 ] ]The function Intersection computes the intersection of relative ideals of a numerical semigroup.
gap> IS = Intersection(Irred);
trueReferences
https://gap-packages.github.io/ numericalsgps.