Apéry set of an integer
Definition
Let \(S\) be a numerical semigroup and \(x \in \mathbb{N} \setminus \{0\}\). It is defined the Apéry set of \(x\) in \(S\), denoted by \(Ap(S,x)\), as
\[ Ap(S,x) = \{s \in S ~ | ~ s - x \not \in S \}. \]
This definition generalizes the definition of Apéry set of \(s \in S \setminus \{0\}\) in \(S\).
Examples
\(\circ\) Let \(S = \langle 3, 8, 13 \rangle = \{0, 3, 6, 8, 9, 11, \rightarrow\}\) and \(x = 2 \not \in S\). The conductor of \(S\) is \(C(S) = 11\), then if \(s \ge C(S) + 2 = 13\), it holds \(s - 2 \ge C(S)\) and \(s-2 \in S\). Therefore, \(Ap(S,2) \subseteq \{0, 3, 6, 8, 9, 11, 12\}\). It is obtained that \(Ap(S,2) = \{0, 3, 6, 9, 12\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 9, 14, 29, 44 \rangle\), in GAP:
gap> S := NumericalSemigroup(9, 14, 29, 44);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an integer \(x \in \mathbb{N} \setminus \{0\}\), the functions AperyList and AperyListOfNumericalSemigroupWRTInteger return the Apéry set of \(x\) in \(S\).
gap> 15 in S;
false
gap> AperyList(S, 15);
[ 0, 9, 14, 18, 23, 27, 28, 32, 36, 37, 41, 45, 46, 50, 54, 55, 63, 64 ]
gap> AperyList(S, 15) = AperyListOfNumericalSemigroupWRTInteger(S, 15);
trueReferences
https://gap-packages.github.io/ numericalsgps.