Division between elements in a numerical semigroup
Definition
Let \(S\) be a numerical semigroup, let \(s, s' \in S\) and \(\le_S\) the relation order of \(S\). It is said that \(s\) divides \(s'\) if \(s \le_S s'\). Equivalently, \(s\) divides \(s'\) if \(s' \in s + S = \{s + x ~ | ~ x \in S\}\). The set of elements \(s\) that divides \(s'\) is the set of divisors of \(s'\) in \(S\) and it is denoted by \(D(s')\).
Examples
\(\circ\) Let \(S = \langle 3, 7, 11 \rangle = \{0, 3, 6, 7, 9, \rightarrow \}\) and \(s' = 12\), let us compute the elements that divides \(s'\). If \(s \in S\) divides \(s'\), then \(s \le s'\), therefore the candidates are \(\{0, 3, 6, 7, 9, 10, 11, 12\}\) and the set of divisors is
\[ D(s') = \{0, 3, 6, 9, 12\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 21, 28, 32, 46 \rangle\), in GAP:
gap> S := NumericalSemigroup(21, 28, 32, 46);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n\) of it, the function DivisorsOfElementInNumericalSemigroup computes the set of divisors of \(n\) in \(S\).
gap> DivisorsOfElementInNumericalSemigroup(S,210);
[ 0, 21, 28, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147,
154, 161, 168, 182, 189, 210 ]
gap> DivisorsOfElementInNumericalSemigroup(210, S);
[ 0, 21, 28, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147,
154, 161, 168, 182, 189, 210 ]References
https://gap-packages.github.io/ numericalsgps.