Minimal set of generators
Definition
Let \(S\) be a numerical semigroup and let \(A \subset S\). We say that \(A\) is a minimal set of generators of \(S\) if \(S = \langle A \rangle\) and no proper subset of \(A\) has this property. It is well known that every numerical semigroup has a unique minimal set of generators, which is \(A = S^* \setminus (S^* + S^*)\). The set of minimal generators is denoted by \(P(S)\) or \(B(S)\), and the elements are called minimal generators or primitive elements.
Examples
\(\circ\) Let \(S = \langle 4, 7, 12, 23 \rangle\). We have \(12 = 4 + 4 + 4\) and \(23 = 4 + 7 + 12\), so \(S = \langle 4, 7, 12, 23 \rangle = \langle 4, 7 \rangle\) and no proper subset of \(B = \{4,7\}\) generates \(S\). Therefore, \(B\) is the set of minimal generators.
\(\circ\) Let \(S = \langle 8, 9, 12, 15 \rangle\) and \(B = \{8, 9, 12, 15\}\). If we remove an element of \(B\), this cannot be generated, and as a consequence, no proper subset of \(B\) generates \(S\). In conclusion, \(B\) is already a minimal set of generators.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 33, 42, 47, 48, 75, 89 \rangle\), in GAP:
gap> S := NumericalSemigroup(15, 33, 42, 47, 48, 75, 89);
<Numerical semigroup with 7 generators>Note that \(B = \{15, 33, 42, 47, 48, 75, 89\}\) is not a minimal set of generators, for example, \(48 = 15 + 33\). The functions MinimalGenerators, MinimalGeneratingSystemOfNumericalSemigroup and MinimalGeneratingSystem compute the minimal set of generators of the numerical semigroup.
gap> MinimalGenerators(S);
[ 15, 33, 42, 47 ]
gap> MinimalGeneratingSystemOfNumericalSemigroup(S);
[ 15, 33, 42, 47 ]
gap> MinimalGeneratingSystem(S);
[ 15, 33, 42, 47 ]If we just want to know what system of generators has been used to generate \(S\), the functions Generators and GeneratorsOfNumericalSemigroup return the set of generators.
gap> Generators(S);
[ 15, 33, 42, 47, 48, 75, 89 ]
gap> GeneratorsOfNumericalSemigroup(S);
[ 15, 33, 42, 47, 48, 75, 89 ]References
https://gap-packages.github.io/ numericalsgps.