Factorizations R-related
Definition
Let \(S\) be a numerical semigroup, an element \(s \in S\), \(\mathbf{Z}(s)\) the set of factorizations of \(s\) and \(\nabla_s\) the graph associated to \(s\). It is said that two factorizations \(a\) and \(b\) of \(s\) are \(\mathcal{R}-\)related if they belong to the same connected component of \(\nabla_s\), that is, there exists a chain of factorizations \(a_1, a_2, \ldots, a_t \in \mathbf{Z}(s)\) such that
\(a_1 = a, a_t = b\).
for all \(i \in \{1, 2, \ldots, t-1\}\), \(a_i \cdot a_{i+1} \ne 0\), where the dot product is component by component.
It can be proven that two elements are \(\mathcal{R}-\)related is an equivalence relation. The equivalence classes are called \(\mathcal{R}-\)classes.
Examples
\(\circ\) Let \(S = \langle 7, 12, 15 \rangle\) and \(s = 36\). Taking into account that \((\alpha, \beta, \gamma) \in \mathbb{Z}(36)\) implies \(0 \le \alpha \le 5, 0 \le \beta \le 3\) and \(0 \le \gamma \le 2\), it is deduced the set of factorizations, which is
\[ \mathbf{Z}(36) = \{(0,3,0), (3,0,1)\}, \]
and \((0,3,0) \cdot (3,0,1) = (0 \cdot 3, 3 \cdot 0, 0 \cdot 1) = (0,0,0)\). Then, \((0,3,0)\) and \((3,0,1)\) are not \(\mathcal{R}-\)related.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 21, 22, 41, 42 \rangle\), in GAP:
gap> S := NumericalSemigroup(21, 22, 41, 42);
<Numerical semigroup with 4 generators>Given a set of factorizations of an element \(s \in S\), where \(S\) is a numerical semigroup, the function RClassesOfSetOfFactorizations returns a list with the \(\mathcal{R}-\)classes in \(\mathbf{Z}(s)\). The set of factorizations of an element can be computed with the functions Factorizations and FactorizationsElementWRTNumericalSemigroup.
gap> z := Factorizations(S, 500);
[ [ 6, 17, 0 ], [ 3, 18, 1 ], [ 0, 19, 2 ], [ 16, 0, 4 ], [ 13, 1, 5 ],
[ 10, 2, 6 ], [ 7, 3, 7 ], [ 4, 4, 8 ], [ 1, 5, 9 ] ]
gap> RClassesOfSetOfFactorizations(z);
[ [ [ 0, 19, 2 ], [ 1, 5, 9 ], [ 3, 18, 1 ], [ 4, 4, 8 ], [ 6, 17, 0 ],
[ 7, 3, 7 ], [ 10, 2, 6 ], [ 13, 1, 5 ], [ 16, 0, 4 ] ] ]References
https://gap-packages.github.io/ numericalsgps.