w-primality of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, \ldots, n_p\}\). It is defined the \(\omega-\)primality of \(S\), denoted by \(\omega(S)\), as
\[ \omega(S) = \max \{ \omega(S, n_i) ~ | ~ i \in \{1, \ldots, p \}\}, \]
where \(\omega(S, n_i)\) denotes the w-primality of \(n_i\) in \(S\) for all \(i \in \{1, \ldots, p\}\).
Given a numerical semigroup \(S\), there is a relation between its [Delta set](Delta-set-NS.qmd \(\Delta(S)\), its [Catenary degree](Catenary-degree-NS.qmd \(\mathbf{C}(S)\), its \(\omega-\)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 \(S = \langle 10, 11, 17, 23 \rangle\). Since \(Betti(S) = \{33, 34, 40, 69 \}\), where \(Betti(S)\) denotes the set of Betti elements of \(S\), it is deduced that its catenary degree is \(\mathbf{C}(S) = 6\), then \(\omega(S) \ge 6\). One can also prove that \(\mathbf{t}(S) = 6\), concluding that \(\omega(S) = 6\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 22, 33, 45, 60, 75 \rangle\), in GAP:
gap> S := NumericalSemigroup(22, 33, 45, 60, 75);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the functions OmegaPrimality and OmegaPrimalityOfNumericalSemigroup return the \(\omega-\)primality of \(S\).
gap> OmegaPrimality(S);
9
gap> OmegaPrimality(S) = OmegaPrimalityOfNumericalSemigroup(S);
trueIf we give also an element \(n \in S\) in the function OmegaPrimality, then it computes the \(\omega-\)primality of \(n\) in \(S\).
gap> OmegaPrimality(S, 207);
15
gap> OmegaPrimality(207, S);
15References
https://gap-packages.github.io/ numericalsgps.