Duplication of a numerical semigroup
Definition
Let \(T\) and \(S\) be numerical semigroups, \(E\) a proper ideal of \(S\) and \(b\) a positive odd integer. It is said that \(T\) is a duplication of \(S\) by \(E\) and \(b\) if \(T = 2S \cup (2E + b)\), where for any subset \(X\) of \(\mathbb{Z}\), \(2X\) denotes the set
\[ 2X = \{2x ~ | ~ x \in X\}. \]
The set \(2S \cup (2E + b)\) is not always a numerical semigroup, but it can be proven that \(E + E + b \subseteq S\) is a sufficient condition, where \(E + E + b\) denotes the set
\[ E + E + b = \{e + e' + b ~ | ~ e, e' \in E\}. \]
Examples
\(\circ\) Let \(S = \langle 3, 8, 10 \rangle, ~ T = \langle 6, 13, 16, 20 \rangle\), \(I = \{6\} + S\) and \(b = 1\), let us prove that \(T\) is a duplication of \(S\) by \(I\) and \(b\). Computing the first elements of \(T\), \(2S\) and \(2I + b\),
\[ T = \{0, 6, 12, 13, 16, 18, 19, 20, 22, 24, 25, 26, 28, \rightarrow\}, \]
\[ 2S = 2 \cdot \{0, 3, 6, 8, \rightarrow \} = \{0, 6, 12\} \cup \{2x ~ | ~ x \ge 8\}, \] \[ 2I + b = 2 \cdot \{6,9,12,14, \rightarrow\} + 1 = \{13, 19, 25\} \cup \{2x + 1 ~ | ~ x \ge 14\}, \] where it is deduced that \(T = 2S \cup (2E + b)\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 23, 82, 93, 94, 143 \rangle\), in GAP:
gap> S := NumericalSemigroup(23, 82, 93, 94, 143);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), a proper ideal \(I\), and an odd integer \(b\), the function NumericalDuplication returns the duplication of \(S\) by \(I\) and \(b\) if the condition \(2I + b \subseteq S\) is satisfied.
gap> I := [92, 138] + S;
<Ideal of numerical semigroup>
gap> IsIntegral(I);
true
gap> b := 13;
13
gap> D := NumericalDuplication(S, I, b);
<Numerical semigroup with 6 generators>\(\diamond\) Let \(T = \langle 8, 12, 14, 27 \rangle\), in GAP:
gap> T := NumericalSemigroup(8, 12, 14, 27);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(T\), the function AsNumericalDuplication returns a list \([S, E, b]\) if \(T\) is a duplication of \(S\) by \(E\) and \(b\) if this is possible, and fail otherwise
gap> L := AsNumericalDuplication(T);
[ <Numerical semigroup with 3 generators>, <Ideal of numerical semigroup>,
7 ]
gap> S := L[1];
<Numerical semigroup with 3 generators>
gap> MinimalGenerators(S);
[ 4, 6, 7 ]
gap> I := L[2];
<Ideal of numerical semigroup>
gap> MinimalGenerators(I);
[ 10 ]
gap> b := L[3];
7Then, \(T = 2S \cup (2I + b)\) where \(S = \langle 4, 6, 7 \rangle, I = \{10\} + S\) and \(b = 7\).
References
https://gap-packages.github.io/ numericalsgps.