Irreducible ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a proper ideal of \(S\). It is said that \(E\) is irreducible if it cannot be expressed as finite intersection of proper ideals of \(S\) strictly containing it.
If \(D(x)\) denotes the set of divisors of x in \(S\), it can be proven that a proper ideal \(E\) is irreducible if, and only if, \(E = S \setminus D(x)\) for some \(x \in S\). In particular, if \(E\) is not irreducible, then is the intersection of \(S \setminus D(x)\) with \(x \in S \setminus E\).
Examples
\(\circ\) Any numerical semigroup \(S\) is irreducible as proper ideal of itself.
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow \}\), \(I = \{3, 5\}\) and \(E = I + S\). First, we compute \(S \setminus E\),
\[ E = I + S = (3 + S) \cup (5 + S) = \{3, 6, 8, \rightarrow \} \cup \{5, 8, 10, \rightarrow \} = \{3, 5, 6, 8, \rightarrow \}. \]
Therefore, \(S \setminus E = \{0, 7\}\) and \(E \subseteq (S \setminus D(0)) \cap (S \setminus D(7))\), but \(S \setminus D(0) = S^*\) and \(S \setminus D(7) = E\), concluding that \(E\) is an irreducible proper ideal of \(S\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 9, 11, 14, 20 \rangle, I = \{20, 27, 33\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(9, 11, 14, 20);
<Numerical semigroup with 4 generators>
gap> I := [20, 27, 33];
[ 20, 27, 33 ]
gap> IS := I + S;
<Ideal of numerical semigroup>Given two proper ideals \(I,J\) of the same numerical semigroup such that \(J \subseteq I\), the function Difference returns \(I \setminus J\). In this case, we need to compute \(S \setminus IS\).
gap> Difference(0+S, IS);
[ 0, 9, 11, 14, 18, 22, 23, 25, 28, 32, 37, 39, 46 ]Then, \(IS = \bigcap_{x \in S \setminus IS} (S \setminus D(x))\). It is known that for every proper ideal there exist an irredundant decomposition into proper irreducible proper ideals, considering \(x \in Maximals_{\le_S}(S \setminus E)\), where \(\le_S\) denotes the relation order of \(S\). The function DecomposeIntegralIdealIntoIrreducibles returns a list with the non redundant irreducible components.
gap> DecomposeIntegralIdealIntoIrreducibles(IS);
[ <Ideal of numerical semigroup>, <Ideal of numerical semigroup>, <Ideal of numerical semigroup> ]
gap> List(DecomposeIntegralIdealIntoIrreducibles(IS), MinimalGenerators);
[ [ 9, 14, 33 ], [ 9, 22 ], [ 11, 27 ] ]Turns out that actually \(IS = ( S \setminus D(22) ) \cap ( S \setminus D(39) ) \cap ( S \setminus D(46) )\), let us check it. Given a numerical semigroup \(S\) and an element \(x \in S\), the function DivisorsOfElementInNumericalSemigroup returns \(D(x)\).
gap> D1 := DivisorsOfElementInNumericalSemigroup(22,S);
[ 0, 11, 22 ]
gap> D2 := DivisorsOfElementInNumericalSemigroup(39,S);
[ 0, 11, 14, 25, 28, 39 ]
gap> D3 := DivisorsOfElementInNumericalSemigroup(46,S);
[ 0, 9, 14, 18, 23, 28, 32, 37, 46 ]Once we have \(D(22), D(39)\) and \(D(46)\), the function IdealByDivisorClosedSet allows to compute \(S \setminus D(x)\).
gap> DifD1 := IdealByDivisorClosedSet(D1,S);
<Ideal of numerical semigroup>
gap> DifD2 := IdealByDivisorClosedSet(D2,S);
<Ideal of numerical semigroup>
gap> DifD3 := IdealByDivisorClosedSet(D2,S);
<Ideal of numerical semigroup>
gap> DifD3 := IdealByDivisorClosedSet(D3,S);
<Ideal of numerical semigroup>Finally, the function Intersection computes the intersection of proper ideals of a numerical semigroup.
gap> IS = Intersection(DifD1, DifD2, DifD3);
trueReferences
https://gap-packages.github.io/ numericalsgps.