Catenary degree of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is defined the catenary degree of \(S\), denoted by \(\mathbf{C}(S)\), as
\[ \mathbf{C}(s) = \sup \{\mathbf{C}(s) ~ | ~ s \in S\}, \]
where \(\mathbf{C}(s)\) denotes the catenary degree of s in \(S\). If \(Betti(S)\) denotes the set of Betti elements of \(S\), it can be proven that
\[ \mathbf{C}(S) = \max \{\mathbf{C}(b) ~ | ~ b \in Betti(S)\}. \]
Given a numerical semigroup \(S\), there is a relation between its Delta set \(\Delta(S)\), its catenary degree \(\mathbf{C}(S)\), its w-primality \(\omega(S)\) and its tame degree \(\mathbf{t}(S)\), which is
\[ \max \Delta(S) + 2 \le \mathbf{C}(S) \le \omega(S) \le \mathbf{t}(S). \]
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle\). The set of Betti elements of \(S\) is \(Betti(S) = \{14, 25, 27\}\), and if we compute the catenary degree of each element, it is obtained that \(\mathbf{C}(14) = 2, \mathbf{C}(25) = 5\) and \(\mathbf{C}(27) = 5\), then \(\mathbf{C}(S) = \max \{2, 5, 5\} = 5\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 37, 42, 45, 80 \rangle\), in GAP:
gap> S := NumericalSemigroup(37, 42, 45, 80);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the functions CatenaryDegree and CatenaryDegreeOfNumericalSemigroup return the catenary degree of \(S\).
gap> CatenaryDegree(S);
8
gap> CatenaryDegreeOfNumericalSemigroup(S) = CatenaryDegree(S);
trueIf we also give an element \(n\) of the numerical semigroup in CatenaryDegree, the function returns the catenary degree of \(n\) in \(S\).
gap> CatenaryDegree(S, 400);
6
gap> CatenaryDegree(400, S);
6References
https://gap-packages.github.io/ numericalsgps.