w-primality of an element

Definition

Let \(S\) be a numerical semigroup and let \(s \in S\). It is defined the \(\omega-\)primality of \(s\) in \(S\), denoted by \(\omega(s, S)\), as the least integer \(N\) such that whenever \(s\) divides \(a_1 + \cdots + a_n\) for some \(a_1, \ldots, a_n \in S\), then \(s\) divides \(a_{i_1} + \cdots a_{i_N}\) for some \(\{i_1, \ldots, i_N\} \subseteq \{1, \ldots, n\}\).

From the \(\omega-\)primality of an element it is defined the w-primality of a numerical semigroup.

Examples

\(\circ\) Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, \ldots, n_p\}\) with \(p > 1\). Let \(i \in \{1, \ldots, p\}\) arbitrary but fixed and let us prove that \(\omega(n_i, S) \ne 1\). Let us consider the Apéry set of \(n_i\) in \(S\), \(Ap(S, n_i)\). It is well known that \(|Ap(S,n)| = n\) and \(F(S) + n \in Ap(S, n)\) for all \(n \in S\), where \(F(S)\) denotes the Frobenius number. Then, let us consider \(a_1 \in Ap(S, n_i) \setminus \{0\}\) and \(a_2 = F(S) + n_i\). We have that \(a_1 + a_2 > F(S) + 1\), which means that \(a_1 + a_2 \in S\). Now, \(a_1 - n_i \not \in S\) and \(a_2 - n_2 \not \in S\) by the definition of Apéry set and consequently \(w(S, n_i) > 1\).

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 42, 56, 71, 90, 104 \rangle\), in GAP:

gap> S := NumericalSemigroup(42, 56, 71, 90, 104);
<Numerical semigroup with 5 generators>

Given a numerical semigroup \(S\) and an element \(n\) of it, the functions OmegaPrimality and OmegaPrimalityOfElementInNumericalSemigroup compute the \(\omega-\)primality of \(n\) in \(S\).

gap> OmegaPrimality(419, S);
15
gap> OmegaPrimality(S, 419);
15
gap> OmegaPrimality(419, S) = OmegaPrimalityOfElementInNumericalSemigroup(419, S);
true

If we only give a numerical semigroup \(S\) in the function OmegaPrimality, then it returns the \(\omega-\)primality of \(S\).

gap> OmegaPrimality(S);
9

Given a numerical semigroup \(S\) and a list of integers \(Is\) of \(S\), the function OmegaPrimalityOfElementListInNumericalSemigroup returns a list with the \(w-\)primality of each element in \(Is\).

gap> Is := [174, 175, 183, 300, 354];
[ 174, 175, 183, 300, 354 ]
gap> IsSubset(S,Is);
true
gap> OmegaPrimalityOfElementListInNumericalSemigroup(Is,S);
[ 11, 10, 11, 14, 17 ]

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.