Apéry set
Definition
Let \(S\) be a numerical semigroup and let \(n \in S \setminus \{0\}\). It is defined the Apéry set of \(S\) with respect to \(n\), denoted by \(Ap(S,n)\) or \(S(n)\), to be the set
\[ Ap(S,n) = \{s \in S ~ | ~ s - n \not \in S\}. \]
The knowledge of this set determines completely the numerical semigroup \(S\). \(Ap(S,n)\) always has \(n\) elements and it is easy to obtain the Frobenius number and the genus of \(S\) from this set. This definition could be extended to the Apéry set of an integer or to the Apéry set of a set of integers.
Examples
\(\circ\) Let \(S = \langle 3, 8, 10 \rangle\), or equivalently, \(S = \{0, 3, 6, 8, \rightarrow \}\), let us compute the Apéry set of \(S\) with respect to \(n = 3\). All the elements greater or equal than \(11\) cannot be in the Apéry set. Indeed, the conductor of \(S\) is \(C(S) = 8\) and if \(n \ge 11\), \(n - 3 \ge 11 - 3 = 8\), concluding that \(n-3 \in S\). With the above reasoning, we have that \(Ap(S, 3) \subseteq \{0, 3, 6, 8, 9, 10\}\), and if we check the condition on each element, it is deduced that \(Ap(S,3) = \{0, 8, 10\}\).
\(\circ\) Let \(S = \langle 4, 6, 11, 13 \rangle\) and \(n = 6\). If we compute the first elements of \(S\), it is deduced that \(S = \{0, 4, 6, 8, 10, \rightarrow \}\) and \(C(S) = 10\), then if \(n \ge 16\), it holds \(n - 6 \in S\) and it must be \(Ap(S,6) \subseteq \{0, 4, 6, 8, 10, 11, 12, 13, 14, 15\}\). If we check on each number, it follows that \(Ap(S,6) = \{0, 4, 8, 11, 13, 15\}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Consider \(S = \langle 12, 14, 25, 33 \rangle\), in GAP:
gap> S := NumericalSemigroup(12, 14, 25, 33);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n \in S \setminus \{0\}\), the functions AperyListOfNumericalSemigroupWRTElement, AperyList and AperyListOfNumericalSemigroup return the Apéry set of \(S\) with respect to \(n\). The codes AperyList(S) and AperyListOfNumericalSemigroup(S) return the Apéry set of \(S\) with respect to \(m(S)\), the multiplicity of \(S\). In this case \(m(S) = 12\).
gap> AperyListOfNumericalSemigroupWRTElement(S, 14);
[ 0, 57, 58, 45, 60, 33, 48, 49, 36, 37, 24, 25, 12, 69 ]
gap> AperyList(S, 12);
[ 0, 25, 14, 39, 28, 53, 42, 67, 56, 33, 58, 47 ]
gap> AperyList(S);
[ 0, 25, 14, 39, 28, 53, 42, 67, 56, 33, 58, 47 ]
brk> AperyListOfNumericalSemigroup(S);
[ 0, 25, 14, 39, 28, 53, 42, 67, 56, 33, 58, 47 ]\(\diamond\) Let \(S = \langle 6,7,8,9 \rangle\). We first compute the left elements and the conductor of \(S\) with the function SmallElements.
gap> S := NumericalSemigroup(6,7,8,9);
<Numerical semigroup with 4 generators>
gap> SmallElements(S);
[ 0, 6, 7, 8, 9, 12 ]We have that \(S = \{0, 6, 7, 8, 9, 12, \rightarrow \}\). Now we calculate the Apéry set of \(S\) with \(n = 6\).
gap> AperyListOfNumericalSemigroupWRTElement(S, 6);
[ 0, 7, 8, 9, 16, 17 ]The Apéry set determines completely the numerical semigroup. The function NumericalSemigroupByAperyList returns the numerical semigroup obtained from a given Apéry set.
gap> T := NumericalSemigroupByAperyList([ 0, 7, 8, 9, 16, 17 ]);
<Numerical semigroup>
gap> SmallElements(T);
[ 0, 6, 7, 8, 9, 12 ]Thus, \(S = \{0, 6, 7, 8, 9, 12, \rightarrow \} = T\) and the result holds. The function IsAperyListOfNumericalSemigroup returns true or false depending on whether a list of integers may represent the Apéry set of a numerical semigroup.
gap> IsAperyListOfNumericalSemigroup([ 0, 7, 8, 9, 16, 17 ]);
trueReferences
https://gap-packages.github.io/ numericalsgps.