Quotient of a numerical semigroup by an integer
Definition
Let \(S\) be a numerical semigroup and let \(p\) be a positive integer. It is defined the quotient of \(S\) by \(p\), denoted by \(\frac{S}{p}\), as
\[ \frac{S}{p} = \{ x \in \mathbb{N} ~ | ~ px \in S \}. \]
It can be proven that \(\frac{S}{p}\) is a numerical semigroup such that \(S \subseteq \frac{S}{p}\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(p = 2\). If \(2x \ge 14\), then \(2x \in S\), therefore \(x \in \frac{S}{2}\) for \(x \ge 7\). If \(x \in \{5, 6\}\), then \(2x \in \{10, 12\} \subseteq S\). Finally, if \(x \in \{1, 2, 3, 4\}\), it is obtained that \(2x \not \in S\). Therefore, \(\frac{S}{2} = \{0, 5, \rightarrow \}\).
\(\circ\) Let \(S = \langle 10, 12, 13 \rangle\) and \(p = 3\). Since \(3 \cdot 10 = 30 = 10 \cdot 3 \in S\), \(3 \cdot 11 = 33 = 10 \cdot 2 + 13 \cdot 1 \in S\) and \(3 \cdot 12 = 12 \cdot 3 \in S\), we have that \(\{10, \rightarrow\} \subseteq \frac{S}{3}\). On the other hand, considering \(x \in \{0, 1, \ldots, 9\}\), it holds that \(3x \in S\) if, and only if, \(x \in \{0, 4, 8\}\), concluding that
\[ \frac{S}{3} = \{0, 4, 8, 10, \rightarrow\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 17, 21, 40, 58 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 17, 21, 40, 58);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\) and a positive integer \(p\), the function QuotientOfNumericalSemigroup returns the numerical semigroup \(\frac{S}{p}\).
gap> S_2 := QuotientOfNumericalSemigroup(S, 2);
<Numerical semigroup>
gap> MinimalGenerators(S_2);
[ 13, 15, 17, 19, 20, 21, 29 ]The function MinimalGenerators returns the minimal generators of the given numerical semigroup, then \(\frac{S}{2} = \langle 13, 15, 17, 19, 20, 21, 29 \rangle\). It is also possible to generate it with the \(/\) operator.
gap> T := S/2;
<Numerical semigroup>
gap> S_2 = T;
trueReferences
https://gap-packages.github.io/ numericalsgps.