Set of divisors
Definition
Let \(S\) be a numerical semigroup and an element \(x \in S\). It is defined the set of divisors of \(x\) in \(S\), denoted by \(D(x)\), as the set
\[ D(x) = \{s \in S ~ | ~ s \le_S x\}, \]
where \(\le_S\) is the relation order of \(S\). This set is always non-empty, since \(x \in D(x)\), and it is related with the notion of proper ideals of a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow \}\) and \(x = 11\). Then, \(s \in D(11)\) if, and only if, \(11 - s \in S\). Checking the condition on each \(s \in S\) such that \(s \le 11\), it is obtained that \(D(11) = \{0, 3, 5, 6, 8, 11\}\).
\(\circ\) Let \(S = \langle 4, 9, 10, 11 \rangle = \{0, 4, 8, \rightarrow \}\). For \(x = 4\), it is obtained \(D(4) = \{0, 4\}\), and for \(x = 8\), the set of divisors is \(D(8) = \{0, 4, 8\}\), then \(D(4) \subseteq D(8)\). In general, given \(x, y \in S\), it is deduced that \(D(x) \subseteq D(y)\) if, and only if, \(x \le_S y\). As \(\le_S\) is a partially relation order, the Hasse diagram can be constructed and given \(x \in S\), \(D(x)\) is the set formed by the elements \(y \in S\) such that there is a path from \(y\) to \(x\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 21, 30, 45, 47 \rangle\), in GAP:
gap> S := NumericalSemigroup(21, 30, 45, 47);
<Numerical semigroup with 4 generators>Let us compute \(D(107)\) (note that \(107 = 2 \cdot 30 + 47 \in S\)). The function DivisorsOfElementInNumericalSemigroup returns the set \(D(x)\) given a numerical semigroup \(S\) and an element \(x \in S\).
gap> DivisorsOfElementInNumericalSemigroup(S, 107);
[ 0, 30, 47, 60, 77, 107 ]References
https://gap-packages.github.io/ numericalsgps.