Arf-irreducible
Definition
Let \(S\) be an Arf numerical semigroup. It is said that \(S\) is Arf-irreducible if \(S\) is irreducible in the Frobenius variety of Arf numerical semigroups.
Examples
\(\circ\) Let us consider the Arf semigroup \(S = \langle 2, 9 \rangle\). The Arf oversemigroups of \(S\), other than \(S\) and \(\mathbb{N}\), are
\[ S_1 = \langle 2, 3 \rangle, ~~ S_2 = \langle 2, 5 \rangle, ~~ S_3 = \langle 2, 9 \rangle. \]
Since \(S_3 \subseteq S_2 \subseteq S_1\), all the intersections generates \(S_1, S_2\) or \(S_3\). Therefore, \(S\) is Arf-irreducible.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 9, 21, 22, 23, 24, 25, 26, 28, 29 \rangle\), in GAP:
gap> S := NumericalSemigroup(9, 21, 22, 23, 24, 25, 26, 28, 29);
<Numerical semigroup with 9 generators>Given a numerical semigroup \(S\), the function IsArf returns true if \(S\) is an Arf semigroup and false otherwise.
gap> IsArf(S);
trueGiven a numerical semigroup \(S\), the function IsArfIrreducible returns true if \(S\) is Arf-irreducible and false otherwise.
gap> IsArfIrreducible(S);
falseIn order to obtain the decomposition into Arf-irreducible, we can apply the function DecomposeIntoArfIrreducibles.
gap> A := DecomposeIntoArfIrreducibles(S);
[ <Numerical semigroup with 3 generators>,
<Numerical semigroup with 9 generators> ]
gap> List(A, l -> MinimalGenerators(l));
[ [ 3, 22, 23 ], [ 9, 16, 20, 21, 22, 23, 24, 26, 28 ] ]The function MinimalGenerators returns the minimal set of generators of the given numerical semigroup. Therefore, the decomposition of \(S\) into Arf-irreducible is
\[ S = \langle 3, 22, 23 \rangle \cap \langle 9, 16, 20, 21, 22, 23, 24, 26, 28 \rangle. \]
References
https://gap-packages.github.io/ numericalsgps.