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### C Contributions

#### C.1 Functions implemented by A. Sammartano

 ‣ IsGradedAssociatedRingNumericalSemigroupBuchsbaum( S ) ( function )

S is a numerical semigroup.

Returns true if the graded ring associated to K[[S]] is Buchsbaum, and false otherwise. This test is the implementation of the algorithm given in [DMV09].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
true


##### C.1-2 IsMpureNumericalSemigroup
 ‣ IsMpureNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Test for the M-Purity of the numerical semigroup S S. This test is based on [Bry10].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsMpureNumericalSemigroup(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsMpureNumericalSemigroup(s);
true


##### C.1-3 IsPureNumericalSemigroup
 ‣ IsPureNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Test for the purity of the numerical semigroup S S. This test is based on [Bry10].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> IsPureNumericalSemigroup(s);
false
gap> s:=NumericalSemigroup(4,6,11);;
gap> IsPureNumericalSemigroup(s);
true


 ‣ IsGradedAssociatedRingNumericalSemigroupGorenstein( S ) ( function )

S is a numerical semigroup.

Returns true if the graded ring associated to K[[S]] is Gorenstein, and false otherwise. This test is the implementation of the algorithm given in [DMS11].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
false
gap> s:=NumericalSemigroup(4,6,11);;
true


 ‣ IsGradedAssociatedRingNumericalSemigroupCI( S ) ( function )

S is a numerical semigroup.

Returns true if the Complete Intersection property of the associated graded ring of a numerical semigroup ring associated to K[[S]], and false otherwise. This test is the implementation of the algorithm given in [DMS13b].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
false
gap> s:=NumericalSemigroup(4,6,11);;
true


##### C.1-6 IsAperySetGammaRectangular
 ‣ IsAperySetGammaRectangular( S ) ( function )

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 [DMS13a].

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


##### C.1-7 IsAperySetBetaRectangular
 ‣ IsAperySetBetaRectangular( S ) ( function )

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 [DMS13a].

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


##### C.1-8 IsAperySetAlphaRectangular
 ‣ IsAperySetAlphaRectangular( S ) ( function )

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 [DMS13a].

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


##### C.1-9 TypeSequenceOfNumericalSemigroup
 ‣ TypeSequenceOfNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Computes the type sequence of a numerical semigroup. This test is the implementation of the algorithm given in [BDF97].

gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
gap> TypeSequenceOfNumericalSemigroup(s);
[ 13, 3, 4, 4, 7, 3, 3, 3, 2, 2, 2, 3, 3, 2, 4, 3, 2, 1, 3, 2, 1, 1, 2, 2, 1,
1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,
1, 1, 1 ]
gap> s:=NumericalSemigroup(4,6,11);;
gap> TypeSequenceOfNumericalSemigroup(s);
[ 1, 1, 1, 1, 1, 1, 1 ]

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