RF-matrix of a pseudo-Frobenius number

Definition

Let \(S\) be a numerical semigroup minimally generated by \(P(S) = \{n_1, \ldots, n_e\}\) with \(n_1 < \cdots < n_e\) and let \(f \in PF(S)\) be a pseudo-Frobenius number. By definition of pseudo-Frobenius number, it holds that \(f + n_i \in S\) for all \(i \in \{1, \ldots, e\}\). Then, it is said that a matrix \(A \in \mathcal{M}_{e \times e}(\mathbb{Z})\) is a \(RF-\)matrix of \(f\) in \(S\) if the \(i-\)th row is a factorization of \(f + n_i\) in \(S\) with \(a_{i,i} = -1\) for all \(i \in \{1, \ldots, e\}\).

Examples

\(\circ\) Let \(S = \langle 5, 7, 13 \rangle = \{0, 5, 7, 10, 12, 13, 14, 15, 17, \rightarrow\}\) and \(f = 8 \in PF(S)\). We have that

\[ 8 + 5 = 13 = 0 \cdot 5 + 0 \cdot 7 + 1 \cdot 13, ~~~ 8 + 7 = 15 = 3 \cdot 5 + 0 \cdot 7 + 0 \cdot 13, ~~~ 8 + 13 = 21 = 0 \cdot 5 + 3 \cdot 7 + 0 \cdot 13 \]

and these are the unique factorizations of \(13, 15\) and \(20\), respectively. Then, the unique \(RF-\)matrix of \(f = 8\) in \(S\) is

\[ A = \begin{pmatrix} -1 & 0 & 1 \\ 3 & -1 & 0 \\ 0 & 3 & -1 \\ \end{pmatrix}. \]

Examples with GAP

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

\(\diamond\) Let \(S = \langle 31, 35, 50, 87, 111, 121, 130 \rangle\), in GAP:

gap> S := NumericalSemigroup(31, 35, 50, 87, 111, 121, 130);
<Numerical semigroup with 7 generators>

Given a numerical semigroup \(S\) and a pseudo-Frobenius number \(f\) of \(S\), the function RFMatrices returns a list with all the \(RF-\)matrices of \(f\) in \(S\). The function PseudoFrobenius computes the set of pseudo-Frobenius numbers of \(S\).

gap> F := PseudoFrobenius(S);
[ 115, 169, 176, 179, 189, 195 ]
gap> RFMatrices(115, S);
[ [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 6, 0, 1, 0, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 6, 0, 1, 0, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 1, 3, 2, 0, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 1, 3, 2, 0, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 2, 0, 0, 2, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 2, 4, 0, -1, 0, 0, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 2, 0, 0, 2, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 6, 0, 1, 0, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 6, 0, 1, 0, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 1, 3, 2, 0, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 1, 3, 2, 0, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 2, 0, 0, 2, 0, -1, 0 ], [ 0, 7, 0, 0, 0, 0, -1 ] ],
  [ [ -1, 1, 0, 0, 1, 0, 0 ], [ 0, -1, 3, 0, 0, 0, 0 ], [ 0, 1, -1, 0, 0, 0, 1 ],
      [ 1, 0, 1, -1, 0, 1, 0 ], [ 0, 3, 0, 0, -1, 1, 0 ],
      [ 2, 0, 0, 2, 0, -1, 0 ], [ 4, 0, 0, 0, 0, 1, -1 ] ] ]

References

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.

Moscariello, Alessio. 2016. “On the Type of an Almost Gorenstein Monomial Curve.” Journal of Algebra 456: 266–77. https://doi.org/https://doi.org/10.1016/j.jalgebra.2016.02.019.