AN ALGORITHM FOR A CLASS OF (n,j,k)- GOOD MATRICES RELATED TO NUMERICAL SEMIGROUPS WITH EMBEDDING DIMENSION 4

Abstract

In this paper, first we recall the definitions of \((n,j)\)-good \(2 \times 2\) and \((n,j,k)\)-good \(3 \times 3\) integer matrices, connected to numerical semigroups of embedding dimension 3 and 4, respectively. Then, for given natural numbers \(n\), \(j\) and \(k\) where \(1 < j, k < n\) and \(k \neq j\), we present an algorithm for obtaining all the \((n,j,k)\)-good matrices
\[
M =
\begin{bmatrix}
a & -u & -p \\
-b & v & -q \\
-c & -w & r
\end{bmatrix}
\]
corresponding to a given \((n,j)\)-good \(2 \times 2\) matrix
\[
K_0 =
\begin{bmatrix}
a_0 & -u_0 \\
-b_0 & v_0
\end{bmatrix}
\]
such that \(a \leq a_0\) and \(v \leq v_0\).