Tame degree of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, n_2, \ldots, n_p\}\). It is defined the tame degree of \(S\), denoted by \(\mathbf{t}(S)\), as
\[ \mathbf{t}(S) = \sup_{n \in S} \mathbf{t}(n), \]
where \(\mathbf{t}(n)\) denotes the tame degree of n in \(S\). If it is defined for all \(i \in \{1, 2, \ldots, p\}\) the set
\[ \mathbf{t}_i(S) = \sup_{n \in S} \mathbf{t}_i(n), \]
it can be proven that \(\mathbf{t}_i(S) = \sup \{\mathbf{t}_i(S) ~ | ~ i \in \{1, 2, \ldots, p\}\}\). Moreover, if for all \(i \in \{1, \ldots, p\}\), it is defined \(S_i\) as the set of \(s \in S\) such that \(\mathbb{Z}(s) \cap Minimals_{\le_S}(\mathbb{Z}(n_i + S))\) is not empty, where \(\mathbb{Z}(n)\) denotes the set of factorizations of \(n\) in \(S\) and \(\le_S\) denotes the relation
\[ (x_1, \ldots, x_p) \le (y_1, \ldots, y_p) \Longleftrightarrow x_i \le y_i ~~ \text{for all } i \in \{1, \ldots, p\}, \] it can be proven that
\[ \mathbf{t}_i(S) = \max \{\mathbf{t}_i(s) ~ | ~ s \in S_i\}. \]
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 \(S = \langle 3, 5, 7 \rangle\). By definition, since \(S_i\) is finite, there exists \(n \in S\) such that \(\mathbf{t}(S) = \mathbf{t}(n)\). Now, it is known that if \(n\) is minimal satisfying the previous equality, then \(G_n\) is not complete, where \(G_n\) denotes the graph associated to n by generators. In this case, for \(n \ge 5 + 7 = 12\) the graph \(G_n\) is complete, then the candidates are in \(B = \{3, 5, 6, 7, 8, 9, 10, 11, 12\}\). Computing the tame degree of each element in \(B\), it is deduced that \(\mathbf{t}(S) = \mathbf{t}(12) = 4\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 37, 42, 45, 65, 88 \rangle\), in GAP:
gap> S := NumericalSemigroup(37, 42, 45, 65, 88);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the functions TameDegree and TameDegreeOfNumericalSemigroup return the tame degree of \(S\).
gap> TameDegree(S);
7
gap> TameDegree(S) = TameDegreeOfNumericalSemigroup(S);
trueIf we also give an element \(n \in S\) in TameDegree, then it computes the tame degree of \(n\) in \(S\).
gap> TameDegree(314, S);
4
gap> TameDegree(S, 314);
4References
https://gap-packages.github.io/ numericalsgps.