Equal catenary degree of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is defined the equal catenary degree of \(S\), denoted by \(EqC(S)\), as
\[ EqC(S) = \max_{s \in S} EqC(s), \]
where \(EqC(s)\) denotes the equal catenary degree of \(s\) in \(S\).
It can be proven that \(EqC(S) = \max_{s \in I} EqC(s)\), where \(I\) is the set of degrees of primitive elements of \(S\).
Examples
Let \(S = \langle 4, 6, 11 \rangle\). The primitive elements of \(S\) are the minimal solutions of the equation
\[ 4(x_1 - y_1) + 6(x_2 - y_2) + 11(x_3 - y_3) = 0. \]
Applying elimination theorem and extension theorem from ring theory in several indeterminate, it is obtained that
\[ M = \langle (1,0,0,1,0,0), (0,1,0,0,1,0), (0, 0, 2, 1, 3, 0), (0, 0, 2, 4, 1, 0), (0, 0, 4, 11, 0, 0), (0, 0, 6, 0, 11, 0), \\ \]
\[ (0, 1, 2, 7, 0, 0), (0, 2, 0, 3, 0, 0), (0, 5, 0, 2, 0, 2), (0, 8, 0, 1, 0, 4), (0,0,1,0,0,1) \rangle, \]
and the degree of primitive elements of \(S\) is \(B =\{4, 6, 11, 12, 22, 28, 30, 44, 48, 66\}\), and \(Eq(S) = \max_{s \in B} ~ Eq(s)\). Computing the set of factorizations of each element in \(B\), it is deduced that \(Eq(S) = 7\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 3, 8, 10 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 8, 10);
<Numerical semigroup with 3 generators>Given a numerical semigroup \(S\), the function EqualCatenaryDegreeOfNumericalSemigroup returns the equal catenary degree of \(S\).
gap> EqualCatenaryDegreeOfNumericalSemigroup(S);
7References
https://gap-packages.github.io/ numericalsgps.