Set closed under divisors

Definition

Let \(S\) be a numerical semigroup, \(\le_S\) the relation order of S and \(X\) a subset of \(\mathbb{N}\). It is said that \(X\) is closed under divisors if for all \(x \in X\) and \(y \in S\) such that \(y \le_S x\), it holds \(y \in X\).

Let \(X\) a subset of \(\mathbb{N}\). It can be proven that \(X\) is a set closed under divisors if, and only if, \(E = S \setminus X\) is a proper ideal of \(S\).

Examples

\(\circ\) Let us prove that if \(E\) is a proper ideal of \(S\), then its complement \(X = S \setminus E\) is closed under divisors. Let \(x \in X\) and \(y \in S\) such that \(y \le_S x\). Therefore, there exists \(s \in S\) so that \(x = y + s\). If we suppose \(y \not \in X\), then \(y \in E\) and \(x = s + y \in S + E \subseteq E\), which is a contradiction. In conclusion, \(y \in X\) and \(X\) is closed under divisors.

\(\circ\) Let \(S = \langle 5, 11, 14 \rangle = \{0, 5, 10, 11, 14, 15, 16, 19, 20, 21, 22, 24 , \rightarrow\}\) and \(E = \{5, 14\} + S \subseteq S\). We have that

\[ E = I + S = \{5, 10, 14, 15, 16, 19, 20, 21, 24 , \rightarrow\}, \]

and \(X = S \setminus E = \{0, 11, 22\}\) is a set closed under divisors.

Examples with GAP

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

\(\diamond\) Let \(S = \langle 10, 11, 15, 19 \rangle\), \(I = \{20, 21, 25\}\) and the proper ideal \(IS = I + S\), in GAP:

gap> S := NumericalSemigroup(10, 11, 15, 19);
<Numerical semigroup with 4 generators>
gap> I := [20, 21, 25];
[ 20, 21, 25 ]
gap> IS := I + S;
<Ideal of numerical semigroup>

Given a subset \(X\) of \(\mathbb{N}\) and a numerical semigroup \(S\), the function IsComplementOfIntegralIdeal returns true or false depending on whether \(X\) is a set closed under divisors or not. The function Difference gets two proper ideals \(I, J\) of the same numerical semigroup with \(J \subseteq I\), and returns \(I \setminus J\).

gap> d := Difference(0 + S, IS);
[ 0, 10, 11, 15, 19, 22, 26, 29, 33, 34, 37, 38, 48 ]
gap> IsComplementOfIntegralIdeal(d, S);
true

Given a set \(X\) closed under divisor, the function IdealByDivisorClosedSet returns the proper ideal \(S \setminus X\).

gap> IS = IdealByDivisorClosedSet(d,S);
true

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.