Apéry table
Definition
Let \(S\) be a numerical semigroup with multiplicity \(m\) and \(M = S \setminus \{0\}\) its maximal ideal with reduction number \(r\). It is defined the Apéry table of \(S\), denoted by \(Ap(S)\), as the matrix \((r+1) \times m\) such that the first row is the Apéry set of \(S\) in \(m\) and the \(k-\)th row is the Apéry set of \((k+1)M\) in \(m\) for \(k \in \{2, \ldots, r\}\), where
\[ kM = \{m_1 + \cdot m_k ~ | ~ m_1, \ldots, m_k \in M\}. \]
The Apéry table can be written as
\[ Ap(S) = \begin{pmatrix} Ap(S, m) \\ Ap(M, m) \\ Ap(2M, m) \\ \vdots \\ Ap(rM, m) \\ \end{pmatrix}, \] where \(Ap(S,m), Ap(M,m), \ldots, Ap(rM,m)\) denotes the Apéry set of \(S, M, 2M, \ldots, rM\) in \(m\), respectively.
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle = \{0, 4, 6, 8, 10, 11, 12, 14, \rightarrow\}\). With few computations, it is obtained that \(r(M) = 2\), where \(r(M)\) denotes the reduction number of \(M\). Also,
\[ \begin{align*} M + M &= \{4, 6, 8, 10, 11, 12, 14, \rightarrow\} + \{4, 6, 8, 10, 11, 12, 14, \rightarrow\} \\ & = (\{4, 6, 8, 10, 11, 12\} + \{4, 6, 8, 10, 11, 12\}) \cup \{18, \rightarrow\} = \{8, 10, 12, 14, \rightarrow\}. \end{align*} \]
Since \(m(S) = 4\), the Apéry set of \(S, M\) and \(2M\) are, respectively,
\[ Ap(S, 4) = \{0, 17, 6, 11\}, ~~ Ap(M, 4) = \{4, 17, 6, 11\}, ~~ Ap(2M, 4) = \{8, 17, 10, 15\}, \] concluding that the Apéry table of \(S\) is as follows.
\[ Ap(S) = \begin{pmatrix} 0 & 17 & 6 & 11 \\ 4 & 17 & 6 & 11 \\ 8 & 17 & 10 & 15 \end{pmatrix}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 20, 50, 54, 78, 115, 116, 119 \rangle\), in GAP:
gap> S := NumericalSemigroup(20, 50, 54, 78, 115, 116, 119);
<Numerical semigroup with 7 generators>Given a numerical semigroup \(S\), the functions AperyTable and AperyTableOfNumericalSemigroup return the Apéry table of \(S\).
gap> AperyTable(S);
[ [ 0, 281, 162, 223, 104, 165, 166, 227, 108, 169, 50, 231, 132, 173, 54, 115,
116, 197, 78, 119 ],
[ 20, 281, 162, 223, 104, 165, 166, 227, 108, 169, 50, 231, 132, 173, 54, 115,
116, 197, 78, 119 ],
[ 40, 281, 162, 223, 104, 165, 166, 227, 108, 169, 70, 231, 132, 173, 74, 135,
136, 197, 98, 139 ],
[ 60, 281, 162, 223, 124, 185, 186, 227, 128, 189, 90, 251, 152, 193, 94, 155,
156, 217, 118, 159 ],
[ 80, 281, 182, 243, 144, 205, 206, 247, 148, 209, 110, 271, 172, 213, 114,
175, 176, 237, 138, 179 ] ]
gap> AperyTableOfNumericalSemigroup(S) = AperyTable(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.