Monotone catenary degree of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is defined the monotone catenary degree of \(S\), denoted by \(MonC(S)\), as
\[ MonC(S) = \max_{s \in S} MonC(s), \]
where \(MonC(s)\) denotes the monotone catenary degree of \(s\) in \(S\).
It can be proven that \(MonC(S) = \max_{s \in I} MonC(S)\), where \(I\) is the set of monotone degrees of primitive elements of \(S\).
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle\). Let \(M\) be the set of solutions of the equation
\[ \left \{ \begin{array}{c} 4(x_1 - y_1) + 6(x_2 - y_2) + 11(x_3 - y_3) = 0, \\ x_1 + x_2 + x_3 - y_1 - y_2 - y_3 + t = 0. \end{array} \right. \]
Applying elimination theorem and extension theorem from ring theory in several indeterminate, it is obtained that the monotone degrees of primitive elements are \(B = \{4, 6, 11, 12, 22, 28, 30, 42, 44, 48, 66 \}\). Computing the set of factorizations of each element in \(B\), it is concluded that \(MonC(S) = 7\).
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 MonotoneCatenaryDegreeOfNumericalSemigroup returns the monotone catenary degree of \(S\).
gap> MonotoneCatenaryDegreeOfNumericalSemigroup(S);
7References
https://gap-packages.github.io/ numericalsgps.