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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 [DMS13].

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 [DMS14].

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 [DMS14].

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 [DMS14].

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 ]


#### C.2 Functions implemented by C. O'Neill

This section includes the implementations of some procedures described in [BOP14].

##### C.2-1 OmegaPrimalityOfElementListInNumericalSemigroup
 ‣ OmegaPrimalityOfElementListInNumericalSemigroup( l, S ) ( function )

S is a numerical semigroup and l a list of elements of S.

Computes the omega-values of all the elements in l.

gap> s:=NumericalSemigroup(10,11,13);;
gap> l:=FirstElementsOfNumericalSemigroup(100,s);;
gap> List(l,x->OmegaPrimalityOfElementInNumericalSemigroup(x,s)); time;
[ 0, 4, 5, 5, 4, 6, 7, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 8, 7, 8, 9, 8, 8, 8,
8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 10, 10, 10, 10, 10,
10, 10, 10, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 12, 12, 12, 12,
12, 12, 12, 12, 13, 14, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14,
14, 14, 14, 14, 14, 15, 16, 15, 15, 15, 15, 15, 15, 15, 15 ]
218
gap> OmegaPrimalityOfElementListInNumericalSemigroup(l,s);time;
[ 0, 4, 5, 5, 4, 6, 7, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 8, 7, 8, 9, 8, 8, 8,
8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 10, 10, 10, 10, 10,
10, 10, 10, 11, 12, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 12, 12, 12, 12,
12, 12, 12, 12, 13, 14, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 14, 14, 14,
14, 14, 14, 14, 14, 15, 16, 15, 15, 15, 15, 15, 15, 15, 15 ]
10


##### C.2-2 FactorizationsElementListWRTNumericalSemigroup
 ‣ FactorizationsElementListWRTNumericalSemigroup( l, S ) ( function )

S is a numerical semigroup and l a list of elements of S.

Computes the factorizations of all the elements in l.

gap> s:=NumericalSemigroup(10,11,13);
<Numerical semigroup with 3 generators>
gap> FactorizationsElementListWRTNumericalSemigroup([100,101,103],s);
[ [ [ 0, 2, 6 ], [ 1, 7, 1 ], [ 3, 4, 2 ], [ 5, 1, 3 ], [ 10, 0, 0 ] ],
[ [ 0, 8, 1 ], [ 1, 0, 7 ], [ 2, 5, 2 ], [ 4, 2, 3 ], [ 9, 1, 0 ] ],
[ [ 0, 7, 2 ], [ 2, 4, 3 ], [ 4, 1, 4 ], [ 7, 3, 0 ], [ 9, 0, 1 ] ] ]



##### C.2-3 DeltaSetPeriodicityBoundForNumericalSemigroup
 ‣ DeltaSetPeriodicityBoundForNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Computes the bound were the periodicity starts for Delta sets of the elements in S; see [GMV14].

gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetPeriodicityBoundForNumericalSemigroup(s);
60


##### C.2-4 DeltaSetPeriodicityStartForNumericalSemigroup
 ‣ DeltaSetPeriodicityStartForNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Computes the element were the periodicity starts for Delta sets of the elements in S.

gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetPeriodicityStartForNumericalSemigroup(s);
21


##### C.2-5 DeltaSetListUpToElementWRTNumericalSemigroup
 ‣ DeltaSetListUpToElementWRTNumericalSemigroup( n, S ) ( function )

S is a numerical semigroup, n a nonnegative integer.

Computes the Delta sets of the integers up to (and including) n, if an integer is not in S, the corresponding Delta set is empty.

gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetListUpToElementWRTNumericalSemigroup(31,s);
[ [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ],
[  ], [  ], [  ], [  ], [  ], [  ], [  ], [  ], [ 2 ], [  ], [  ], [ 2 ], [  ],
[ 2 ], [  ], [ 2 ], [ 2 ], [  ] ]


##### C.2-6 DeltaSetUnionUpToElementWRTNumericalSemigroup
 ‣ DeltaSetUnionUpToElementWRTNumericalSemigroup( n, S ) ( function )

S is a numerical semigroup, n a nonnegative integer.

Computes the union of the delta sets of the elements of S up to and including n, using a ring buffer to conserve memory.

gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetUnionUpToElementWRTNumericalSemigroup(60,s);
[ 2 ]


##### C.2-7 DeltaSetOfNumericalSemigroup
 ‣ DeltaSetOfNumericalSemigroup( S ) ( function )

S is a numerical semigroup.

Computes the Delta set of S.

gap> s:=NumericalSemigroup(5,7,11);;
gap> DeltaSetOfNumericalSemigroup(s);
[ 2 ]

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