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### 7. Ideals of numerical semigroups

#### 7.1 Ideals of numerical semigroups

Let S be a numerical semigroup. A set I of integers is an ideal relative to a numerical semigroup S provided that I+Ssubseteq I and that there exists din S such that d+Isubseteq S.

If i_1,...,i_k is a subset of Z, then the set I=i_1,...,i_k+S=bigcup_n=1^k i_n+S is an ideal relative to S, and i_1,..., i_k is a system of generators of I. A system of generators M is minimal if no proper subset of M generates the same ideal. Usually, ideals are specified by means of its generators and the ambient numerical semigroup to which they are ideals (for more information see for instance [VF97]).

##### 7.1-1 IdealOfNumericalSemigroup

`> IdealOfNumericalSemigroup`( l, S ) | ( function )

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

The output is the ideal of S generated by l

There are several shortcuts for this function, as shown in the example.

    gap> IdealOfNumericalSemigroup([3,5],NumericalSemigroup(9,11));
    <Ideal of numerical semigroup>
    gap> [3,5]+NumericalSemigroup(9,11);
    <Ideal of numerical semigroup>
    gap> last=last2;
    true
    gap> 3+NumericalSemigroup(5,9);
    <Ideal of numerical semigroup>

##### 7.1-2 IsIdealOfNumericalSemigroup

`> IsIdealOfNumericalSemigroup`( Obj ) | ( function )

Tests if the object Obj is an ideal of a numerical semigroup.

    gap> I:=[1..7]+NumericalSemigroup(7,19);
    <Ideal of numerical semigroup>
    gap> IsIdealOfNumericalSemigroup(I);
    true
    gap> IsIdealOfNumericalSemigroup(2);
    false

##### 7.1-3 MinimalGeneratingSystemOfIdealOfNumericalSemigroup

`> MinimalGeneratingSystemOfIdealOfNumericalSemigroup`( I ) | ( function )

I is an ideal of a numerical semigroup.

The output is the minimal system of generators of I.

    gap> I:=[3,5,9]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
    [ 3 ]

##### 7.1-4 GeneratorsOfIdealOfNumericalSemigroup

`> GeneratorsOfIdealOfNumericalSemigroup`( I ) | ( function )

`> GeneratorsOfIdealOfNumericalSemigroupNC`( I ) | ( function )

I is an ideal of a numerical semigroup.

The output of `GeneratorsOfIdealOfNumericalSemigroup` is a system of generators of the ideal. If the minimal system of generators is known, then it is used as output. `GeneratorsOfIdealOfNumericalSemigroupNC` always returns the set of generators stored in I!.generators.

    gap> I:=[3,5,9]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> GeneratorsOfIdealOfNumericalSemigroup(I);
    [ 3, 5, 9 ]
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
    [ 3 ]
    gap> GeneratorsOfIdealOfNumericalSemigroup(I);
    [ 3 ]
    gap> GeneratorsOfIdealOfNumericalSemigroupNC(I);
    [ 3, 5, 9 ]

##### 7.1-5 AmbientNumericalSemigroupOfIdeal

`> AmbientNumericalSemigroupOfIdeal`( I ) | ( function )

I is an ideal of a numerical semigroup, say S.

The output is S.

    gap> I:=[3,5,9]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> AmbientNumericalSemigroupOfIdeal(I);
    <Numerical semigroup with 2 generators>

##### 7.1-6 SmallElementsOfIdealOfNumericalSemigroup

`> SmallElementsOfIdealOfNumericalSemigroup`( I ) | ( function )

I is an ideal of a numerical semigroup.

The output is a list with the elements in I that are less than or equal to the greatest integer not belonging to the ideal plus one.

    gap> I:=[3,5,9]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> SmallElementsOfIdealOfNumericalSemigroup(I);
    [ 3, 5, 7, 9, 11, 13 ]
    gap> J:=[2,11]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> SmallElementsOfIdealOfNumericalSemigroup(J);
    [ 2, 4, 6, 8, 10 ]

##### 7.1-7 BelongsToIdealOfNumericalSemigroup

`> BelongsToIdealOfNumericalSemigroup`( n, I ) | ( function )

I is an ideal of a numerical semigroup, n is an integer.

The output is true if n belongs to I.

n in I can be used for short.

    gap> J:=[2,11]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> BelongsToIdealOfNumericalSemigroup(9,J);
    false
    gap> 9 in J;
    false
    gap> BelongsToIdealOfNumericalSemigroup(10,J);
    true
    gap> 10 in J;
    true

##### 7.1-8 SumIdealsOfNumericalSemigroup

`> SumIdealsOfNumericalSemigroup`( I, J ) | ( function )

I, J are ideals of a numerical semigroup.

The output is the sum of both ideals i+j | iin I, jin J.

I + J is a synonym of this function.

    gap> I:=[3,5,9]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> J:=[2,11]+NumericalSemigroup(2,11);
    <Ideal of numerical semigroup>
    gap> I+J;
    <Ideal of numerical semigroup>
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
    [ 5, 14 ]
    gap> SumIdealsOfNumericalSemigroup(I,J);
    <Ideal of numerical semigroup>
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);
    [ 5, 14 ]

##### 7.1-9 MultipleOfIdealOfNumericalSemigroup

`> MultipleOfIdealOfNumericalSemigroup`( n, I ) | ( function )

I is an ideal of a numerical semigroup, n is a non negative integer.

The output is the ideal I+cdots+I (n times).

n * I can be used for short.

    gap> I:=[0,1]+NumericalSemigroup(3,5,7);
    <Ideal of numerical semigroup>
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(2*I);
    [ 0, 1, 2 ]

##### 7.1-10 SubtractIdealsOfNumericalSemigroup

`> SubtractIdealsOfNumericalSemigroup`( I, J ) | ( function )

I, J are ideals of a numerical semigroup.

The output is the ideal zin Z | z+Jsubseteq I.

I - J is a synonym of this function. The following example appears in [HS04].

    gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);
    <Numerical semigroup with 5 generators>
    gap> I:=[0,1]+S;
    <Ideal of numerical semigroup>
    gap> II:=(0+S)-I;
    <Ideal of numerical semigroup>
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);
    [ 0, 1 ]
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(II);
    [ 14, 20 ]
    gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I+II);
    [ 14, 15, 20, 21 ]

##### 7.1-11 DifferenceOfIdealsOfNumericalSemigroup

`> DifferenceOfIdealsOfNumericalSemigroup`( I, J ) | ( function )

I, J are ideals of a numerical semigroup. J must be contained in I.

The output is the set I\ J.

    gap> S:=NumericalSemigroup(14, 15, 20, 21, 25);
    <Numerical semigroup with 5 generators>
    gap> I:=[0,1]+S;
    <Ideal of numerical semigroup>
    gap> 2*I-2*I;
    <Ideal of numerical semigroup>
    gap> I-I;
    <Ideal of numerical semigroup>
    gap> DifferenceOfIdealsOfNumericalSemigroup(last2,last);
    [ 26, 27, 37, 38 ]

##### 7.1-12 HilbertFunctionOfIdealOfNumericalSemigroup

`> HilbertFunctionOfIdealOfNumericalSemigroup`( n, I ) | ( function )

I is an ideal of a numerical semigroup, n is a non negative integer. I must be contained in its ambient semigroup.

The output is the cardinality of the set nI\ (n+1)I.

    gap> I:=[6,9,11]+NumericalSemigroup(6,9,11);;
    gap> List([1..7],n->HilbertFunctionOfIdealOfNumericalSemigroup(n,I));
    [ 3, 11, 24, 48, 96, 192, 384 ]

##### 7.1-13 BlowUpIdealOfNumericalSemigroup

`> BlowUpIdealOfNumericalSemigroup`( I ) | ( function )

I is an ideal of a numerical semigroup.

The output is the ideal bigcup_n>= 0 nI-nI.

    gap> I:=[0,2]+NumericalSemigroup(6,9,11);;
    gap> BlowUpIdealOfNumericalSemigroup(I);;
    gap> SmallElementsOfIdealOfNumericalSemigroup(last);
    [ 0, 2, 4, 6, 8 ]

##### 7.1-14 MaximalIdealOfNumericalSemigroup

`> MaximalIdealOfNumericalSemigroup`( S ) | ( function )

Returns the maximal ideal of the numerical semigroup S.

    gap> MaximalIdealOfNumericalSemigroup(NumericalSemigroup(3,7));
    <Ideal of numerical semigroup>

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