Complete intersection numerical semigroups The cardinality of a minimal presentation of a numerical semigroup is alwas greater than or equal to its embedding dimension minus one. Complete intersection numerical semigroups are numerical semigroups reching this bound, and they are irreducible. It can be shown that every complete intersection (other that \mathbb N) is a complete intersection if and only if it is the gluing of two complete intersections. When in this gluing, one of the copies is isomorphic to \mathbb N, then we obtain a free semigroup in the sense of . Two special kinds of free semigroups are telescopic semigroups () and those associated to an irreducible planar curve (). We use the algorithms presented in to find the set of all complete intersections (also free, telescopic and associated to irreducible planar curves) numerical semigroups with given Frobenius number.

s is a numerical semigroup. Returns all partitions \{A_1,A_2\} of the minimal generating set of s such that s is a gluing of \langle A_1\rangle and \langle A_2\rangle by gcd(A_1)gcd(A_2)
gap> s := NumericalSemigroup( 10, 15, 16 ); <Numerical semigroup with 3 generators> gap> AsGluingOfNumericalSemigroups(s); [ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ] gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 ); <Numerical semigroup with 8 generators> gap> AsGluingOfNumericalSemigroups(s); [ ]
s is a numerical semigroup. The output is true if the numerical semigroup is a complete intersection, that is, the cardinality of a (any) minimal presentation equals its embedding dimension minus one.
gap> s := NumericalSemigroup( 10, 15, 16 ); <Numerical semigroup with 3 generators> gap> IsACompleteIntersectionNumericalSemigroup(s); true gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 ); <Numerical semigroup with 8 generators> gap> IsACompleteIntersectionNumericalSemigroup(s); false
f is an integer greater than or equal to -1. The output is the set of all complete intersection numerical semigroups with frobenius number f.
gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57)); 34
s is a numerical semigroup. The output is true if the numerical semigroup is free in the sense of : it is either \mathbb N or the gluing of a copy of \mathbb N with a free numerical semigroup.
gap> IsFreeNumericalSemigroup(NumericalSemigroup(10,15,16)); true gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7)); false
f is an integer greater than or equal to -1. The output is the set of all free numerical semigroups with frobenius number f.
gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57)); 33
s is a numerical semigroup. The output is true if the numerical semigroup is telescopic in the sense of : it is either \mathbb N or the gluing of \langle n_e\rangle and s'=\langle n_1/d,\ldots, n_{e-1}/d\rangle, and s' is again a telescopic numerical semigroup, where n_1 < \cdots < n_e are the minimal generators of s.
gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14)); false gap> IsFreeNumericalSemigroup(NumericalSemigroup(4,11,14)); true
f is an integer greater than or equal to -1. The output is the set of all telescopic numerical semigroups with frobenius number f.
gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57)); 20
s is a numerical semigroup. The output is true if the numerical semigroup is associated to an irreducible planar curve singularity (). These semigroups are telescopic.
gap> ns := NumericalSemigroup(4,11,14);; gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns); false gap> ns := NumericalSemigroup(4,11,19);; gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns); true
f is an integer greater than or equal to -1. The output is the set of all numerical semigroups associated to irreducible planar curves singularities with frobenius number f.
gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57)); 7
S is a numerical semigroup.

Test for the \gamma-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in . Numerical Semigroups with this property are free and thus complete intersections.
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);; gap> IsAperySetGammaRectangular(s); false gap> s:=NumericalSemigroup(4,6,11);; gap> IsAperySetGammaRectangular(s); true
S is a numerical semigroup.

Test for the \beta-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in ; \beta-rectangularity implies \gamma-rectangularity.
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);; gap> IsAperySetBetaRectangular(s); false gap> s:=NumericalSemigroup(4,6,11);; gap> IsAperySetBetaRectangular(s); true
S is a numerical semigroup.

Test for the \alpha-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in ; \alpha-rectangularity implies \beta-rectangularity.
gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);; gap> IsAperySetAlphaRectangular(s); false gap> s:=NumericalSemigroup(4,6,11);; gap> IsAperySetAlphaRectangular(s); true