Shaded set
Definition
Let \(S\) be a numerical semigroup minimally generated by \(A = \{n_1, n_2, \ldots, n_p\}\) and let \(n \in S\). it is defined the shaded set of \(n\) in \(S\), denoted by \(\mathcal{F}(n)\), as
\[ \mathcal{F}(n) = \left \{ V \subseteq A ~ | ~ n - \sum_{a \in V} a \in S \right \}. \]
It can be proven that \(A \in \mathcal{F}(n)\) if, and only if, \(\mathcal{F}(n) = \mathcal{P}(A)\), where \(\mathcal{P}(A)\) denotes the partition set of \(A\).
Examples
\(\circ\) Let \(S = \langle 5, 9, 13 \rangle\) and \(n = 35 \in S\). In order to compute the shaded set of \(n\) in \(S\), we divide in cases:
\(V \subseteq A\) has \(0\) elements: we have \(V = \emptyset\) and \(V \in \mathcal{F}(35)\).
\(V \subseteq A\) has \(1\) element: the candidates are \(T_{1,1} = \{5\}, T_{1,2} = \{9\}\) and \(T_{1,3} = \{13\}\). Since \(35 - 5 = 30, 35 - 9 = 26, 35 - 13 = 22\) and \(\{30, 26, 22\} \subseteq S\), it is obtained that \(T_{1,1}, T_{1,2}, T_{1,3} \in \mathcal{F}(35)\).
\(V \subseteq A\) has \(2\) element: the candidates are \(T_{2,1} = \{5,9\}, T_{2,2} = \{5, 13\}\) and \(T_{2,3} = \{9, 13\}\), but \(T_{2,3}\) is the only one that satisfies the condition. Thus,\(T_{2,3} \in \mathcal{F}(35)\) and \(T_{2,1}, T_{2,2} \not \in \mathcal{F}(35)\).
\(V \subseteq A\) has \(3\) element: we have \(V = A\) and \(35 - (5 + 9 + 13) = 8 \not \in S\), then \(A \not \in \mathcal{F}(35)\).
To sum up, the shaded set of \(n = 35\) in \(S\) is
\[ \mathcal{F}(35) = \{\emptyset, \{5\}, \{9\}, \{13\}, \{9, 13\}\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 20, 27, 31, 32 \rangle\), in GAP:
gap> S := NumericalSemigroup(15, 20, 27, 31, 32);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\) and an element \(n\) of it, the function ShadedSetOfElementInNumericalSemigroup returns the shaded set of \(n\) in \(S\).
gap> ShadedSetOfElementInNumericalSemigroup(85,S);
[ [ ], [ 15 ], [ 15, 20 ], [ 20 ], [ 27 ], [ 27, 31 ], [ 31 ] ]There may be \(n, m \in S\) such that \(\mathcal{F}(n) = \mathcal{F}(m)\) and \(n \ne m\).
gap> ShadedSetOfElementInNumericalSemigroup(70,S);
[ [ ], [ 15 ], [ 15, 20 ], [ 20 ] ]
gap> ShadedSetOfElementInNumericalSemigroup(75,S);
[ [ ], [ 15 ], [ 15, 20 ], [ 20 ] ]
gap> ShadedSetOfElementInNumericalSemigroup(80,S);
[ [ ], [ 15 ], [ 15, 20 ], [ 20 ] ]References
https://gap-packages.github.io/ numericalsgps.