  
  [1X12 [33X[0;0YGood semigroups[133X[101X
  
  [33X[0;0YWe will only cover here good semigroups of [22XN^2[122X.[133X
  
  [33X[0;0YA good semigroup [22XS[122X is a submonoid of [22XN^2[122X, with the following properties.[133X
  
  [33X[0;0Y(G1) It is closed under infimums (minimum componentwise).[133X
  
  [33X[0;0Y(G2)  if [22Xa, b ∈ M[122X and [22Xa_i = b_i[122X for some [22Xi ∈ {1, 2}[122X, then there exists [22Xc ∈ M[122X
  such that [22Xc_i > a_i = b_i[122X and [22Xc_j = min{a_j,b_j}[122X, with [22Xj∈{1,2}∖ {i}[122X.[133X
  
  [33X[0;0Y(G3) there exists [22XC∈N^n[122X such that [22XC+N^n⊆ S[122X.[133X
  
  [33X[0;0YValue  semigroups  of  algebroid branches are good semigroups, but there are
  good  semigroups that are not of this form. Since good semigroups are closed
  under  infimums, if [22XC_1[122X and [22XC_2[122X fulfill [22XC_i+N^n⊆ S[122X, then [22XC_1∧ C_2+N^n⊆ S[122X. So
  there  is  a minimum [22XC[122X fulfilling [22XC+N^n⊆ S[122X, which is called the conductor of
  [22XS[122X.[133X
  
  [33X[0;0YThe contents of this chapter are described in [DGSM16].[133X
  
  
  [1X12.1 [33X[0;0YDefining good semigroups[133X[101X
  
  [1X12.1-1 IsGoodSemigroup[101X
  
  [29X[2XIsGoodSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0YDetects if [3XS[103X is an object of type good semigroup.[133X
  
  [1X12.1-2 NumericalSemigroupDuplication[101X
  
  [29X[2XNumericalSemigroupDuplication[102X( [3XS[103X, [3XE[103X ) [32X function
  
  [33X[0;0Y[3XS[103X  is  a numerical semigroup and [3XE[103X is an ideal of [3XS[103X with [22XE⊆ S[122X. The output is
  [22XS⋈ E= D∪ (E× E)∪{ a∧ b∣ a∈ D, b∈ E× E},[122X where [22XD={(s,s)∣ s∈ S}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xl:=Cartesian([1..11],[1..11]);;[127X[104X
    [4X[25Xgap>[125X [27XIntersection(dup,l);[127X[104X
    [4X[28X[ [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],[128X[104X
    [4X[28X  [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ],[128X[104X
    [4X[28X  [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ],[128X[104X
    [4X[28X  [ 11, 9 ], [ 11, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27X[384938749837,349823749827] in dup;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.1-3 AmalgamationOfNumericalSemigroups[101X
  
  [29X[2XAmalgamationOfNumericalSemigroups[102X( [3XS[103X, [3XE[103X, [3Xb[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a numerical semigroup, [3XE[103X is an ideal of a numerical semigroup [22XT[122X with [22XE⊆
  T[122X, and [3Xb[103X is an integer such that multiplication by [3Xb[103X is a morphism from [3XS[103X to
  [22XT[122X, say [22Xg[122X. The output is [22XS⋈^g E= D∪(g^-1(E)× E)∪ {a∧ b∣ a∈ D, b∈ g^-1(E)× E},[122X
  where [22XD={(s,b s)∣ s∈ S}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(2,3);;[127X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroup(3,4);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=3+t;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=AmalgamationOfNumericalSemigroups(s,e,2);;[127X[104X
    [4X[25Xgap>[125X [27X[2,3] in dup;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.1-4 CartesianProductOfNumericalSemigroups[101X
  
  [29X[2XCartesianProductOfNumericalSemigroups[102X( [3XS[103X, [3XT[103X ) [32X function
  
  [33X[0;0Y[3XS[103X  and  [3XT[103X  are  numerical  semigroups.  The  output is [22XS× T[122X, which is a good
  semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(2,3);;[127X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroup(3,4);;[127X[104X
    [4X[25Xgap>[125X [27XIsGoodSemigroup(CartesianProductOfNumericalSemigroups(s,t));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.1-5 GoodSemigroup[101X
  
  [29X[2XGoodSemigroup[102X( [3XX[103X, [3XC[103X ) [32X function
  
  [33X[0;0Y[3XX[103X  is  a list of points with nonnegative integer coordinates and [22XC[122X is a pair
  of  nonnegative  integer  (a list with two elements). If [22XM[122X is the affine and
  infimum  closure of [3XX[103X, decides if it is a good semigroup, and if so, outputs
  it.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];[127X[104X
    [4X[28X[ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],[128X[104X
    [4X[28X  [ 25, 12 ], [ 25, 16 ] ][128X[104X
    [4X[25Xgap>[125X [27XC:=[25,27];[127X[104X
    [4X[28X[ 25, 27 ][128X[104X
    [4X[25Xgap>[125X [27XGoodSemigroup(G,C);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
  [4X[32X[104X
  
  
  [1X12.2 [33X[0;0YNotable elements[133X[101X
  
  [1X12.2-1 BelongsToGoodSemigroup[101X
  
  [29X[2XBelongsToGoodSemigroup[102X( [3Xv[103X, [3XS[103X ) [32X operation
  [29X[2X\in[102X( [3Xv[103X, [3XS[103X ) [32X operation
  
  [33X[0;0Y[3XS[103X  is  a good semigroup and [3Xv[103X is a pair of integers. The output is [10Xtrue[110X if [3Xv[103X
  is  in [3XS[103X, and [10Xfalse[110X otherwise. Other ways to use this operation are [10X\in(v,S)[110X
  and [10Xv in S[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(2,3);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27XBelongsToGoodSemigroup([2,2],dup);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X[2,2] in dup;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X[3,2] in dup;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X12.2-2 Conductor[101X
  
  [29X[2XConductor[102X( [3XS[103X ) [32X function
  [29X[2XConductorOfGoodSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a good semigroup. The output is its conductor.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XConductor(dup);[127X[104X
    [4X[28X[ 11, 11 ][128X[104X
    [4X[25Xgap>[125X [27XConductorOfGoodSemigroup(dup);[127X[104X
    [4X[28X[ 11, 11 ][128X[104X
  [4X[32X[104X
  
  [1X12.2-3 SmallElements[101X
  
  [29X[2XSmallElements[102X( [3XS[103X ) [32X function
  [29X[2XSmallElementsOfGoodSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a good semigroup. The output is its set of small elements, that is, the
  elements  smaller  than  its  conductor  with  respect  to the usual partial
  ordering.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElementsOfGoodSemigroup(dup);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ],[128X[104X
    [4X[28X  [ 6, 10 ], [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ],[128X[104X
    [4X[28X  [ 9, 9 ], [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ],[128X[104X
    [4X[28X  [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.2-4 RepresentsSmallElementsOfGoodSemigroup[101X
  
  [29X[2XRepresentsSmallElementsOfGoodSemigroup[102X( [3XX[103X ) [32X function
  
  [33X[0;0Y[3XX[103X  is  a list of points in the nonnegative orthant of the plane with integer
  coordinates. Determines if it represents the set of small elements of a good
  semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElementsOfGoodSemigroup(dup);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],[128X[104X
    [4X[28X  [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],[128X[104X
    [4X[28X  [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentsSmallElementsOfGoodSemigroup(last);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.2-5 GoodSemigroupBySmallElements[101X
  
  [29X[2XGoodSemigroupBySmallElements[102X( [3XX[103X ) [32X function
  
  [33X[0;0Y[3XX[103X  is  a list of points in the nonnegative orthant of the plane with integer
  coordinates. Determines if it represents the set of small elements of a good
  semigroup,  and  then  outputs  the  good semigroup having [3XX[103X as set of small
  elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElementsOfGoodSemigroup(dup);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],[128X[104X
    [4X[28X  [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],[128X[104X
    [4X[28X  [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27XG:=GoodSemigroupBySmallElements(last);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xdup=G;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.2-6 MaximalElementsOfGoodSemigroup[101X
  
  [29X[2XMaximalElementsOfGoodSemigroup[102X( [3XS[103X ) [32X attribute
  
  [33X[0;0Y[3XS[103X is a good semigroup. The output is the set of elements [22X(x,y)[122X of [3XS[103X with the
  following  property:  there  is  no  other element [22X(x',y')[122X in [3XS[103X with [22X(x,y)le
  (x',y')[122X sharing a coordinate with [22X(x,y)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[4,3],[7,13],[11,17]];;[127X[104X
    [4X[25Xgap>[125X [27Xg:=GoodSemigroup(G,[11,17]);;[127X[104X
    [4X[25Xgap>[125X [27Xmx:=MaximalElementsOfGoodSemigroup(g);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 4, 3 ], [ 7, 13 ], [ 8, 6 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.2-7 IrreducibleMaximalElementsOfGoodSemigroup[101X
  
  [29X[2XIrreducibleMaximalElementsOfGoodSemigroup[102X( [3XS[103X ) [32X attribute
  
  [33X[0;0Y[3XS[103X  is  a  good  semigroup. The output is the set of elements nonzero maximal
  elements  that  cannot be expressed as a sum of two nonzero maximal elements
  of the good semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[4,3],[7,13],[11,17]];;[127X[104X
    [4X[25Xgap>[125X [27Xg:=GoodSemigroup(G,[11,17]);;[127X[104X
    [4X[25Xgap>[125X [27XIrreducibleMaximalElementsOfGoodSemigroup(g);[127X[104X
    [4X[28X[ [ 4, 3 ], [ 7, 13 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.2-8 GoodSemigroupByMaximalElements[101X
  
  [29X[2XGoodSemigroupByMaximalElements[102X( [3XS[103X, [3XT[103X, [3XM[103X, [3XC[103X ) [32X function
  
  [33X[0;0Y[3XS[103X  and  [3XT[103X  are  numerical semigroups, [3XM[103X is a list of pairs in [22XS× T[122X. [3XC[103X is the
  conductor, and thus a pair of nonnegative integers. The output is the set of
  elements  of [22XS× T[122X that are not above an element in [3XM[103X, that is, if they share
  a  coordinate  with  an  element in [3XM[103X, then they must be smaller or equal to
  that  element  with  respect  to the usual partial ordering. The output is a
  good semigroup, if [3XM[103X is an correct set of maximal elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[4,3],[7,13],[11,17]];;[127X[104X
    [4X[25Xgap>[125X [27Xg:=GoodSemigroup(G,[11,17]);;[127X[104X
    [4X[25Xgap>[125X [27Xsm:=SmallElements(g);;[127X[104X
    [4X[25Xgap>[125X [27Xmx:=MaximalElementsOfGoodSemigroup(g);;[127X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroupBySmallElements(Set(sm,x->x[1]));;[127X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroupBySmallElements(Set(sm,x->x[2]));;[127X[104X
    [4X[25Xgap>[125X [27XConductor(g);[127X[104X
    [4X[28X[ 11, 15 ][128X[104X
    [4X[25Xgap>[125X [27Xgg:=GoodSemigroupByMaximalElements(s,t,mx,[11,15]);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xgg=g;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.2-9 MinimalGoodGeneratingSystemOfGoodSemigroup[101X
  
  [29X[2XMinimalGoodGeneratingSystemOfGoodSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X  is  a  good  semigroup.  The output is its minimal good generating system
  (which is unique in the local case).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGoodGeneratingSystemOfGoodSemigroup(dup);[127X[104X
    [4X[28X[ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.2-10 MinimalGenerators[101X
  
  [29X[2XMinimalGenerators[102X( [3XS[103X ) [32X attribute
  
  [33X[0;0YThis is just a synonym of [10X MinimalGoodGeneratingSystemOfGoodSemigroup[110X ([10XS[110X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=6+s;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGenerators(dup);[127X[104X
    [4X[28X[ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X12.3 [33X[0;0YSymmetric semigroups[133X[101X
  
  [1X12.3-1 IsSymmetricGoodSemigroup[101X
  
  [29X[2XIsSymmetricGoodSemigroup[102X( [3XS[103X ) [32X attribute
  [29X[2XIsSymmetric[102X( [3XS[103X ) [32X attribute
  
  [33X[0;0Y[3XS[103X is a good semigroup. Determines if [3XS[103X is a symmetric good semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=CanonicalIdealOfNumericalSemigroup(s);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=15+e;;[127X[104X
    [4X[25Xgap>[125X [27Xdup:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27XIsSymmetricGoodSemigroup(dup);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.3-2 ArfGoodSemigroupClosure[101X
  
  [29X[2XArfGoodSemigroupClosure[102X( [3XS[103X ) [32X function
  [29X[2XArfClosure[102X( [3XS[103X ) [32X operation
  
  [33X[0;0Y[3XS[103X is a good semigroup. Determines the Arf good semigroup closure of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[3,3],[4,4],[5,4],[4,6]];[127X[104X
    [4X[28X[ [ 3, 3 ], [ 4, 4 ], [ 5, 4 ], [ 4, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27XC:=[6,6];[127X[104X
    [4X[28X[ 6, 6 ][128X[104X
    [4X[25Xgap>[125X [27XS:=GoodSemigroup(G,C);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElements(S);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ], [ 4, 6 ], [ 5, 4 ], [ 6, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27XA:=ArfGoodSemigroupClosure(S);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElements(A);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X12.4 [33X[0;0YGood ideals[133X[101X
  
  [33X[0;0YA  relative  ideal [22XI[122X of a relative good semigroup [22XM[122X is a relative good ideal
  if [22XI[122X fulfills conditions (G1) and (G2) of the definition of good semigroup.[133X
  
  [1X12.4-1 GoodIdeal[101X
  
  [29X[2XGoodIdeal[102X( [3XX[103X, [3XS[103X ) [32X function
  
  [33X[0;0Y[3XX[103X  is  a  list  of points with nonnegative integer coordinates and [22XS[122X is good
  semigroup.  Decides  if the closure of [22XX+S[122X under infimums is a relative good
  ideal of [22XS[122X, and if so, outputs it.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];[127X[104X
    [4X[28X[ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],[128X[104X
    [4X[28X[ 25, 12 ], [ 25, 16 ] ][128X[104X
    [4X[25Xgap>[125X [27XC:=[25,27];[127X[104X
    [4X[28X[ 25, 27 ][128X[104X
    [4X[25Xgap>[125X [27Xg := GoodSemigroup(G,C);[127X[104X
    [4X[28X<Good semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xi:=GoodIdeal([[2,3]],g);[127X[104X
    [4X[28X<Good ideal of good semigroup>[128X[104X
  [4X[32X[104X
  
  [1X12.4-2 GoodGeneratingSystemOfGoodIdeal[101X
  
  [29X[2XGoodGeneratingSystemOfGoodIdeal[102X( [3XI[103X ) [32X function
  
  [33X[0;0Y[3XI[103X  is  a  good  ideal  of  a good semigroup. The output is a good generating
  system of [3XI[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xd:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=GoodIdeal([[2,3],[3,2],[2,2]],d);;[127X[104X
    [4X[25Xgap>[125X [27XGoodGeneratingSystemOfGoodIdeal(e);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.4-3 AmbientGoodSemigroupOfGoodIdeal[101X
  
  [29X[2XAmbientGoodSemigroupOfGoodIdeal[102X( [3XI[103X ) [32X function
  
  [33X[0;0YIf [3XI[103X is a good ideal of a good semigroup [22XM[122X, then the output is [22XM[122X. The output
  is a good generating system of [3XI[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xa:=AmalgamationOfNumericalSemigroups(s,e,5);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=GoodIdeal([[2,3],[3,2],[2,2]],a);;[127X[104X
    [4X[25Xgap>[125X [27Xa=AmbientGoodSemigroupOfGoodIdeal(e);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.4-4 MinimalGoodGeneratingSystemOfGoodIdeal[101X
  
  [29X[2XMinimalGoodGeneratingSystemOfGoodIdeal[102X( [3XI[103X ) [32X function
  
  [33X[0;0Y[3XI[103X  is  a  good  ideal  of  a  good semigroup. The output is the minimal good
  generating system of [3XI[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xd:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=GoodIdeal([[2,3],[3,2],[2,2]],d);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalGoodGeneratingSystemOfGoodIdeal(e);[127X[104X
    [4X[28X[ [ 2, 3 ], [ 3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.4-5 BelongsToGoodIdeal[101X
  
  [29X[2XBelongsToGoodIdeal[102X( [3Xv[103X, [3XI[103X ) [32X operation
  [29X[2X\in[102X( [3Xv[103X, [3XI[103X ) [32X operation
  
  [33X[0;0Y[3XI[103X  is  a  good  ideal  of  a good semigroup and [3Xv[103X is a pair of integers. The
  output  is  [10Xtrue[110X  if  [3Xv[103X is in [3XI[103X, and [10Xfalse[110X otherwise. Other ways to use this
  operation are [10X\in(v,I)[110X and [10Xv in I[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xd:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=GoodIdeal([[2,3],[3,2]],d);;[127X[104X
    [4X[25Xgap>[125X [27X[1,1] in e;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27X[2,2] in e;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X12.4-6 SmallElementsOfGoodIdeal[101X
  
  [29X[2XSmallElementsOfGoodIdeal[102X( [3XI[103X ) [32X function
  [29X[2XSmallElements[102X( [3XI[103X ) [32X function
  
  [33X[0;0Y[3XI[103X  is  a  good  ideal. The output is its set of small elements, that is, the
  elements  smaller  than  its  conductor  and larger than its minimum element
  (with respect to the usual partial ordering).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xd:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=GoodIdeal([[2,3],[3,2]],d);;[127X[104X
    [4X[25Xgap>[125X [27XSmallElements(e);[127X[104X
    [4X[28X[ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ], [ 5, 5 ], [ 5, 6 ], [ 6, 5 ], [ 7, 7 ] ][128X[104X
  [4X[32X[104X
  
  [1X12.4-7 CanonicalIdealOfGoodSemigroup[101X
  
  [29X[2XCanonicalIdealOfGoodSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a good semigroup. The output is the canonical ideal of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xe:=10+s;;[127X[104X
    [4X[25Xgap>[125X [27Xd:=NumericalSemigroupDuplication(s,e);;[127X[104X
    [4X[25Xgap>[125X [27Xc:=CanonicalIdealOfGoodSemigroup(d);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalGoodGeneratingSystemOfGoodIdeal(c);[127X[104X
    [4X[28X[ [ 0, 0 ], [ 2, 2 ] ][128X[104X
  [4X[32X[104X
  
