[1X7 Ideals of numerical semigroups[0X [1X7.1 Ideals of numerical semigroups[0X Let S be a numerical semigroup. A set I of integers is an [13Xideal relative[0m 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 [BF97]). [1X7.1-1 IdealOfNumericalSemigroup[0m [2X> IdealOfNumericalSemigroup( [0X[3Xl, S[0X[2X ) ________________________________[0Xfunction [3XS[0m is a numerical semigroup and [3Xl[0m a list of integers. The output is the ideal of [3XS[0m generated by [3Xl[0m There are several shortcuts for this function, as shown in the example. [4X--------------------------- Example ----------------------------[0X [4Xgap> IdealOfNumericalSemigroup([3,5],NumericalSemigroup(9,11));[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> [3,5]+NumericalSemigroup(9,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> last=last2;[0X [4Xtrue[0X [4Xgap> 3+NumericalSemigroup(5,9);[0X [4X<Ideal of numerical semigroup>[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-2 IsIdealOfNumericalSemigroup[0m [2X> IsIdealOfNumericalSemigroup( [0X[3XObj[0X[2X ) _______________________________[0Xfunction Tests if the object [3XObj[0m is an ideal of a numerical semigroup. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[1..7]+NumericalSemigroup(7,19);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> IsIdealOfNumericalSemigroup(I);[0X [4Xtrue[0X [4Xgap> IsIdealOfNumericalSemigroup(2);[0X [4Xfalse[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-3 MinimalGeneratingSystemOfIdealOfNumericalSemigroup[0m [2X> MinimalGeneratingSystemOfIdealOfNumericalSemigroup( [0X[3XI[0X[2X ) __________[0Xfunction [3XI[0m is an ideal of a numerical semigroup. The output is the minimal system of generators of [3XI[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[3,5,9]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);[0X [4X[ 3 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-4 GeneratorsOfIdealOfNumericalSemigroup[0m [2X> GeneratorsOfIdealOfNumericalSemigroup( [0X[3XI[0X[2X ) _______________________[0Xfunction [2X> GeneratorsOfIdealOfNumericalSemigroupNC( [0X[3XI[0X[2X ) _____________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup. The output of [10XGeneratorsOfIdealOfNumericalSemigroup[0m is a system of generators of the ideal. If the minimal system of generators is known, then it is used as output. [10XGeneratorsOfIdealOfNumericalSemigroupNC[0m always returns the set of generators stored in [3XI!.generators[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[3,5,9]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> GeneratorsOfIdealOfNumericalSemigroup(I);[0X [4X[ 3, 5, 9 ][0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);[0X [4X[ 3 ][0X [4Xgap> GeneratorsOfIdealOfNumericalSemigroup(I);[0X [4X[ 3 ][0X [4Xgap> GeneratorsOfIdealOfNumericalSemigroupNC(I);[0X [4X[ 3, 5, 9 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-5 AmbientNumericalSemigroupOfIdeal[0m [2X> AmbientNumericalSemigroupOfIdeal( [0X[3XI[0X[2X ) ____________________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup, say S. The output is S. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[3,5,9]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> AmbientNumericalSemigroupOfIdeal(I);[0X [4X<Numerical semigroup with 2 generators>[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-6 SmallElementsOfIdealOfNumericalSemigroup[0m [2X> SmallElementsOfIdealOfNumericalSemigroup( [0X[3XI[0X[2X ) ____________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup. The output is a list with the elements in [3XI[0m that are less than or equal to the greatest integer not belonging to the ideal plus one. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[3,5,9]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> SmallElementsOfIdealOfNumericalSemigroup(I);[0X [4X[ 3, 5, 7, 9, 11, 13 ][0X [4Xgap> J:=[2,11]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> SmallElementsOfIdealOfNumericalSemigroup(J);[0X [4X[ 2, 4, 6, 8, 10 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-7 BelongsToIdealOfNumericalSemigroup[0m [2X> BelongsToIdealOfNumericalSemigroup( [0X[3Xn, I[0X[2X ) _______________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup, [3Xn[0m is an integer. The output is true if [3Xn[0m belongs to [3XI[0m. [3X n in I[0m can be used for short. [4X--------------------------- Example ----------------------------[0X [4Xgap> J:=[2,11]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> BelongsToIdealOfNumericalSemigroup(9,J);[0X [4Xfalse[0X [4Xgap> 9 in J;[0X [4Xfalse[0X [4Xgap> BelongsToIdealOfNumericalSemigroup(10,J);[0X [4Xtrue[0X [4Xgap> 10 in J;[0X [4Xtrue[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-8 SumIdealsOfNumericalSemigroup[0m [2X> SumIdealsOfNumericalSemigroup( [0X[3XI, J[0X[2X ) ____________________________[0Xfunction [3XI, J[0m are ideals of a numerical semigroup. The output is the sum of both ideals { i+j | iin [3XI[0m, jin [3XJ[0m}. [3XI + J[0m is a synonym of this function. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[3,5,9]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> J:=[2,11]+NumericalSemigroup(2,11);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> I+J;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);[0X [4X[ 5, 14 ][0X [4Xgap> SumIdealsOfNumericalSemigroup(I,J);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(last);[0X [4X[ 5, 14 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-9 MultipleOfIdealOfNumericalSemigroup[0m [2X> MultipleOfIdealOfNumericalSemigroup( [0X[3Xn, I[0X[2X ) ______________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup, [3Xn[0m is a non negative integer. The output is the ideal [3XI[0m+cdots+[3XI[0m ([3Xn[0m times). [3X n * I[0m can be used for short. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[0,1]+NumericalSemigroup(3,5,7);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(2*I);[0X [4X[ 0, 1, 2 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-10 SubtractIdealsOfNumericalSemigroup[0m [2X> SubtractIdealsOfNumericalSemigroup( [0X[3XI, J[0X[2X ) _______________________[0Xfunction [3XI, J[0m are ideals of a numerical semigroup. The output is the ideal { zin Z | z+[3XJ[0msubseteq [3XI[0m}. [3XI - J[0m is a synonym of this function. The following example appears in [HS04a]. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=NumericalSemigroup(14, 15, 20, 21, 25);[0X [4X<Numerical semigroup with 5 generators>[0X [4Xgap> I:=[0,1]+S;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> II:=(0+S)-I;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I);[0X [4X[ 0, 1 ][0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(II);[0X [4X[ 14, 20 ][0X [4Xgap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(I+II);[0X [4X[ 14, 15, 20, 21 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-11 DifferenceOfIdealsOfNumericalSemigroup[0m [2X> DifferenceOfIdealsOfNumericalSemigroup( [0X[3XI, J[0X[2X ) ___________________[0Xfunction [3XI, J[0m are ideals of a numerical semigroup. [3XJ[0m must be contained in [3XI[0m. The output is the set [3XI[0m\ [3XJ[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=NumericalSemigroup(14, 15, 20, 21, 25);[0X [4X<Numerical semigroup with 5 generators>[0X [4Xgap> I:=[0,1]+S;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> 2*I-2*I;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> I-I;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> DifferenceOfIdealsOfNumericalSemigroup(last2,last);[0X [4X[ 26, 27, 37, 38 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-12 TranslationOfIdealOfNumericalSemigroup[0m [2X> TranslationOfIdealOfNumericalSemigroup( [0X[3Xk, I[0X[2X ) ___________________[0Xfunction Given an ideal [3XI[0m of a numerical semigroup S and an integer [3Xk[0m returns an ideal of the numerical semigroup S generated by {i_1+k,...,i_n+k} where {i_1,...,i_n} is the system of generators of [3XI[0m. As a synonym to [10XTranslationOfIdealOfNumericalSemigroup(k, I)[0m the expression [10Xk + I[0m may be used. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(13,23);[0X [4X<Numerical semigroup with 2 generators>[0X [4Xgap> l:=List([1..6], _ -> Random([8..34]));[0X [4X[ 22, 29, 34, 25, 10, 12 ][0X [4Xgap> I:=IdealOfNumericalSemigroup(l, s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> It:=TranslationOfIdealOfNumericalSemigroup(7,I);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> It2:=7+I;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> It2=It;[0X [4Xtrue[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-13 HilbertFunctionOfIdealOfNumericalSemigroup[0m [2X> HilbertFunctionOfIdealOfNumericalSemigroup( [0X[3Xn, I[0X[2X ) _______________[0Xfunction [3XI[0m is an ideal of a numerical semigroup, [3Xn[0m is a non negative integer. [3XI[0m must be contained in its ambient semigroup. The output is the cardinality of the set [3Xn[0m[3XI[0m\ ([3Xn[0m+1)[3XI[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[6,9,11]+NumericalSemigroup(6,9,11);;[0X [4Xgap> List([1..7],n->HilbertFunctionOfIdealOfNumericalSemigroup(n,I));[0X [4X[ 3, 5, 6, 6, 6, 6, 6 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-14 BlowUpIdealOfNumericalSemigroup[0m [2X> BlowUpIdealOfNumericalSemigroup( [0X[3XI[0X[2X ) _____________________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup. The output is the ideal bigcup_n>= 0 n[3XI[0m-n[3XI[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[0,2]+NumericalSemigroup(6,9,11);;[0X [4Xgap> BlowUpIdealOfNumericalSemigroup(I);;[0X [4Xgap> SmallElementsOfIdealOfNumericalSemigroup(last);[0X [4X[ 0, 2, 4, 6, 8 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-15 ReductionNumberIdealNumericalSemigroup[0m [2X> ReductionNumberIdealNumericalSemigroup( [0X[3XI[0X[2X ) ______________________[0Xfunction [3XI[0m is an ideal of a numerical semigroup. The output is the least integer such that n [3XI[0m + i=(n+1)[3XI[0m, where i=min([3XI[0m). [4X--------------------------- Example ----------------------------[0X [4Xgap> I:=[0,2]+NumericalSemigroup(6,9,11);;[0X [4Xgap> ReductionNumberIdealNumericalSemigroup(I);[0X [4X2[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-16 MaximalIdealOfNumericalSemigroup[0m [2X> MaximalIdealOfNumericalSemigroup( [0X[3XS[0X[2X ) ____________________________[0Xfunction Returns the maximal ideal of the numerical semigroup [3XS[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> MaximalIdealOfNumericalSemigroup(NumericalSemigroup(3,7));[0X [4X<Ideal of numerical semigroup>[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-17 BlowUpOfNumericalSemigroup[0m [2X> BlowUpOfNumericalSemigroup( [0X[3XS[0X[2X ) __________________________________[0Xfunction If [3XM[0m is the maximal ideal of the numerical semigroup, then the output is the numerical semigroup bigcup_n>= 0 n[3XM[0m-n[3XM[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);[0X [4X<Numerical semigroup with 10 generators>[0X [4Xgap> BlowUpOfNumericalSemigroup(s);[0X [4X<Numerical semigroup with 10 generators>[0X [4Xgap> SmallElementsOfNumericalSemigroup(last);[0X [4X[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,[0X [4X 40, 41, 42, 44 ][0X [4Xgap> m:=MaximalIdealOfNumericalSemigroup(s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> BlowUpIdealOfNumericalSemigroup(m);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> SmallElementsOfIdealOfNumericalSemigroup(last);[0X [4X[ 0, 5, 10, 12, 15, 17, 20, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 37, 39,[0X [4X 40, 41, 42, 44 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-18 MicroInvariantsOfNumericalSemigroup[0m [2X> MicroInvariantsOfNumericalSemigroup( [0X[3XS[0X[2X ) _________________________[0Xfunction Returns the microinvariants of the numerical semigroup [3XS[0m defined in [Eli01]. For their computation we have used the formula given in [BF06]. The Ap\'ery set of [3XS[0m and its blow up are involved in this computation. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);[0X [4X<Numerical semigroup with 10 generators>[0X [4Xgap> bu:=BlowUpOfNumericalSemigroup(s);[0X [4X<Numerical semigroup with 10 generators>[0X [4Xgap> ap:=AperyListOfNumericalSemigroupWRTElement(s,30);;[0X [4Xgap> apbu:=AperyListOfNumericalSemigroupWRTElement(bu,30);;[0X [4Xgap> (ap-apbu)/30;[0X [4X[ 0, 4, 4, 3, 2, 1, 3, 4, 4, 3, 2, 3, 1, 4, 4, 3, 3, 1, 4, 4, 4, 3, 2, 4, 2,[0X [4X 5, 4, 3, 3, 2 ][0X [4Xgap> MicroInvariantsOfNumericalSemigroup(s)=last;[0X [4Xtrue[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-19 IsGradedAssociatedRingNumericalSemigroupCM[0m [2X> IsGradedAssociatedRingNumericalSemigroupCM( [0X[3XS[0X[2X ) __________________[0Xfunction Returns true if the graded ring associated to K[[[3XS[0m]] is Cohen-Macaulay, and false otherwise. This test is the implementation of the algorithm given in [BF06]. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);[0X [4X<Numerical semigroup with 10 generators>[0X [4Xgap> IsGradedAssociatedRingNumericalSemigroupCM(s);[0X [4Xfalse[0X [4Xgap> MicroInvariantsOfNumericalSemigroup(s);[0X [4X[ 0, 4, 4, 3, 2, 1, 3, 4, 4, 3, 2, 3, 1, 4, 4, 3, 3, 1, 4, 4, 4, 3, 2, 4, 2,[0X [4X 5, 4, 3, 3, 2 ][0X [4Xgap> List(AperyListOfNumericalSemigroupWRTElement(s,30),[0X [4X> w->MaximumDegreeOfElementWRTNumericalSemigroup (w,s));[0X [4X[ 0, 1, 4, 1, 2, 1, 3, 1, 4, 3, 2, 3, 1, 1, 4, 3, 3, 1, 4, 1, 4, 3, 2, 4, 2,[0X [4X5, 4, 3, 1, 2 ][0X [4Xgap> last=last2;[0X [4Xfalse[0X [4Xgap> s:=NumericalSemigroup(4,6,11);[0X [4X<Numerical semigroup with 3 generators>[0X [4Xgap> IsGradedAssociatedRingNumericalSemigroupCM(s);[0X [4Xtrue[0X [4Xgap> MicroInvariantsOfNumericalSemigroup(s);[0X [4X[ 0, 2, 1, 1 ][0X [4Xgap> List(AperyListOfNumericalSemigroupWRTElement(s,4),[0X [4X> w->MaximumDegreeOfElementWRTNumericalSemigroup(w,s));[0X [4X[ 0, 2, 1, 1 ][0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-20 CanonicalIdealOfNumericalSemigroup[0m [2X> CanonicalIdealOfNumericalSemigroup( [0X[3XS[0X[2X ) __________________________[0Xfunction Computes a canonical ideal of [3XS[0m ([BF06]): { x in Z | g-x not in S}. [4X--------------------------- Example ----------------------------[0X [4Xgap> s:=NumericalSemigroup(4,6,11);[0X [4X<Numerical semigroup with 3 generators>[0X [4Xgap> m:=MaximalIdealOfNumericalSemigroup(s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> c:=CanonicalIdealOfNumericalSemigroup(s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> (m-c)-c=m;[0X [4Xtrue[0X [4Xgap> id:=3+s;[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> (id-c)-c=id;[0X [4Xtrue[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-21 IntersectionIdealsOfNumericalSemigroup[0m [2X> IntersectionIdealsOfNumericalSemigroup( [0X[3XI, J[0X[2X ) ___________________[0Xfunction Given two ideals [3XI[0m and [3XJ[0m of a numerical semigroup [3XS[0m returns the ideal of the numerical semigroup [3XS[0m which is the intersection of the ideals [3XI[0m and [3XJ[0m. [4X--------------------------- Example ----------------------------[0X [4Xgap> i:=IdealOfNumericalSemigroup([75,89],s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> j:=IdealOfNumericalSemigroup([115,289],s);[0X [4X<Ideal of numerical semigroup>[0X [4Xgap> IntersectionIdealsOfNumericalSemigroup(i,j);[0X [4X<Ideal of numerical semigroup>[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X [1X7.1-22 IsMonomialNumericalSemigroup[0m [2X> IsMonomialNumericalSemigroup( [0X[3XS[0X[2X ) ________________________________[0Xfunction [3XS[0m is a numerical semigroup. Tests whether [3XS[0m a monomial numerical semigroup. Let R a Noetherian ring such that K subseteq R subseteq K[[t]], K is a field of characteristic zero, the algebraic closure of R is K[[t]], and the conductor (R : K[[t]]) is not zero. If v : K((t))-> Z is the natural valuation for K((t)), then v(R) is a numerical semigroup. Let S be a numerical semigroup minimally generated by {n_1,...,n_e}. The semigroup ring associated to S is K[[S]]=K[[t^n_1,...,t^n_e]]. A ring is called a semigroup ring if it is of the form K[[S]], for some numerical semigroup S. We say that S is a monomial numerical semigroup if for any R as above with v(R)=S, R is a semigroup ring. See [VMi02] for details. [4X--------------------------- Example ----------------------------[0X [4Xgap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,7));[0X [4Xtrue[0X [4Xgap> IsMonomialNumericalSemigroup(NumericalSemigroup(4,6,11));[0X [4Xfalse[0X [4X[0X [4X [0X [4X------------------------------------------------------------------[0X