Apéry list of a relative ideal

Definition

Let \(S\) be a numerical semigroup, let \(E\) be a relative ideal of \(S\) and \(n \in S\). It is defined the Apéry set of \(E\) in \(n\), denoted by \(Ap(E, n)\), as

\[ Ap(E, n) = \{w \in E ~ | ~ w - n \not \in E\}. \]

It can be proven that \(Ap(E, n) = \{w(0), w(1), \ldots, w(n-1)\}\), where \(w(i) \in E\) is the least element in \(E\) such that \(w(i) \equiv i ~ (mod ~ n)\). In particular, \(|Ap(E, n)| = n\).

Examples

\(\circ\) Let \(S = \langle 4, 9, 15 \rangle = \{0, 4, 8, 9, 12, 13, 15, \rightarrow \}\), \(I = \{-2, 3, 5\} + S\) and \(n = 4\). If \(L = \{0, 4, 8, 9, 12, 13\}\) and \(L - n = \{l - n ~ | ~ l \in L\}\),

\[ I = (L-2) \cup (L+3) \cup (L+5) \cup \{13, \rightarrow\} = \{-2, 2, 3, 5, 6, 7, 9, \rightarrow\}, \]

and with few computations, the Apéry set of \(I\) in \(n = 4\) is

\[ Ap(I,4) = \{12, 5, -2, 3\}. \]

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 39, 56, 93, 96, 104, 105\rangle\) and \(I = \{47, 57, 70, 92, 100\} + S\), in GAP:

gap> S := NumericalSemigroup(39, 56, 93, 96, 104, 105);
<Numerical semigroup with 6 generators>
gap> I := [47, 57, 70, 92, 100] + S;
<Ideal of numerical semigroup>

Given a relative ideal \(I\) of a numerical semigroup \(S\) and an integer \(n \in S\); the functions AperyList and AperyListOfIdealOfNumericalSemigroupWRTElement return the Apéry set of \(I\) in \(n\).

gap> AperyList(I, 56);
[ 224, 57, 170, 227, 228, 229, 174, 175, 232, 233, 178, 179, 236, 125, 70, 239,
  240, 185, 242, 131, 188, 189, 190, 135, 192, 193, 306, 139, 140, 197, 86, 143,
  200, 201, 202, 203, 92, 205, 150, 151, 96, 153, 266, 267, 100, 213, 214, 47,
  272, 161, 162, 163, 164, 109, 166, 279 ]
gap> AperyListOfIdealOfNumericalSemigroupWRTElement(I, 56) = AperyList(I, 56);
true

If no integer is given, the function AperyList computes the Apéry set of \(I\) in \(m(S)\), where \(m(S)\) denotes the multiplicity of \(S\).

gap> AperyList(I);
[ 156, 196, 197, 159, 199, 161, 162, 163, 47, 126, 166, 206, 207, 169, 92, 249,
  289, 212, 57, 175, 215, 255, 100, 140, 219, 103, 143, 222, 262, 185, 225, 70,
  188, 150, 151, 113, 153, 193, 233 ]
gap> m := Multiplicity(S);
39
gap> AperyList(I, m) = AperyList(I);
true

References

Casabella, D’Anna, L. 2024. “Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup.” Mediterr. J. Math. 21 (7). https://doi.org/10.1007/s00009-023-02550-8.

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.