Desert of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup and \(G(S)\) the set of gaps of \(S\). A subset \(A \subseteq \mathbb{Z}\) is a desert of \(S\) if \(A = \mathbb{Z}^{< 0}\) or \(A \subseteq G(S)\) and \(\min A - 1, \max A + 1 \in S\). Equivalently, \(A\) is a desert of \(S\) if \(A\) is a maximal interval of gaps of \(S\) (counting also the set of all negative integers).
It can be proven that the number of deserts is \(|Ap(S, 1)|\), where \(Ap(S,1)\) denotes the Apéry set of the integer \(1\).
Examples
\(\circ\) Let \(S = \langle 4, 7, 10 \rangle = \{ 0, 4, 7, 8, 10, 11, 12, 14, \rightarrow \}\). The deserts of \(S\) are
\[ A_0 = \mathbb{Z}^{<0}, A_1 = \{1, 2, 3\}, A_2 = \{5, 6\}, A_3 = \{9\}, A_4 = \{13\}. \]
On the other hand,
\[ Ap(S,1) = \{0, 4, 7, 10, 14\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 21, 35, 43, 45, 79 \rangle\), in GAP:
gap> S := NumericalSemigroup(21, 35, 43, 45, 79);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the functions Deserts and DesertsOfNumericalSemigroup return a list with all the deserts of \(S\), not counting the desert formed by the all negative integers.
gap> Deserts(S);
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ],
[ 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 ], [ 36, 37, 38, 39, 40, 41 ],
[ 44 ], [ 46, 47, 48, 49, 50, 51, 52, 53, 54, 55 ], [ 57, 58, 59, 60, 61, 62 ],
[ 65 ], [ 67, 68, 69 ], [ 71, 72, 73, 74, 75, 76 ], [ 81, 82, 83 ], [ 89 ],
[ 92, 93, 94, 95, 96, 97 ], [ 102, 103, 104 ], [ 110 ], [ 116, 117, 118 ],
[ 137, 138, 139 ] ]
gap> Length(Deserts(S)) + 1;
17On the other hand, given a numerical semigroup \(S\) and an integer \(x \in \mathbb{N} \setminus \{0\}\), the function AperyList returns the Apéry set of \(x\) in \(S\).
gap> AperyList(S, 1);
[ 0, 21, 35, 42, 45, 56, 63, 66, 70, 77, 84, 90, 98, 105, 111, 119, 140 ]
gap> Length(AperyList(S,1));
17References
https://gap-packages.github.io/ numericalsgps.