  
  [1X8 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X[101X
  
  
  [1X8.1 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X[101X
  
  [33X[0;0YIf  [22XS[122X is a numerical semigroup and [22Xm[122X is its multiplicity (the least positive
  integer  belonging  to  it),  then  the  embedding  dimension  [22Xe[122X  of  [22XS[122X (the
  cardinality  of the minimal system of generators of [22XS[122X) is less than or equal
  to  [22Xm[122X.  We  say  that [22XS[122X has maximal embedding dimension (MED for short) when
  [22Xe=m[122X. The intersection of two numerical semigroups with the same multiplicity
  and  maximal  embedding  dimension  is again of maximal embedding dimension.
  Thus  we  define  the MED closure of a non-empty subset of positive integers
  [22XM={m  <  m_1  <  ⋯  <  m_n  <⋯}[122X with [22Xgcd(M)=1[122X as the intersection of all MED
  numerical semigroups with multiplicity [22Xm[122X.[133X
  
  [33X[0;0YGiven  a  MED  numerical  semigroup [22XS[122X, we say that [22XM={m_1 < ⋯< m_k}[122X is a MED
  system  of generators if the MED closure of [22XM[122X is [22XS[122X. Moreover, [22XM[122X is a minimal
  MED  generating system for [22XS[122X provided that every proper subset of [22XM[122X is not a
  MED  system  of  generators of [22XS[122X. Minimal MED generating systems are unique,
  and  in  general  are  smaller than the classical minimal generating systems
  (see [RGSGGB03]).[133X
  
  [1X8.1-1 IsMED[101X
  
  [29X[2XIsMED[102X( [3XS[103X ) [32X property
  [29X[2XIsMEDNumericalSemigroup[102X( [3XS[103X ) [32X property
  
  [33X[0;0Y[3XS[103X  is  a numerical semigroup. Returns true if [3XS[103X is a MED numerical semigroup
  and false otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsMEDNumericalSemigroup(NumericalSemigroup(3,5,7));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMEDNumericalSemigroup(NumericalSemigroup(3,5));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X8.1-2 MEDNumericalSemigroupClosure[101X
  
  [29X[2XMEDNumericalSemigroupClosure[102X( [3XS[103X ) [32X function
  [29X[2XMEDClosure[102X( [3XS[103X ) [32X operation
  
  [33X[0;0Y[3XS[103X is a numerical semigroup. Returns the MED closure of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMEDNumericalSemigroupClosure(NumericalSemigroup(3,5));[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGenerators(last);[127X[104X
    [4X[28X[ 3, 5, 7 ][128X[104X
  [4X[32X[104X
  
  [1X8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup[101X
  
  [29X[2XMinimalMEDGeneratingSystemOfMEDNumericalSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a MED numerical semigroup. Returns the minimal MED generating system of
  [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalMEDGeneratingSystemOfMEDNumericalSemigroup([127X[104X
    [4X[25X>[125X [27XNumericalSemigroup(3,5,7));[127X[104X
    [4X[28X[ 3, 5 ][128X[104X
  [4X[32X[104X
  
  
  [1X8.2 [33X[0;0YNumerical semigroups with the Arf property and Arf closures[133X[101X
  
  [33X[0;0YNumerical  semigroups  with the Arf property are a special kind of numerical
  semigroups  with maximal embedding dimension. A numerical semigroup [22XS[122X is Arf
  if for every [22Xx,y,z[122X in [22XS[122X with [22Xx≥ y≥ z[122X, one has that [22Xx+y-z∈ S[122X.[133X
  
  [33X[0;0YThe  intersection  of two Arf numerical semigroups is again Arf, and thus we
  can  consider the Arf closure of a set of nonnegative integers with greatest
  common  divisor  equal to one. Analogously as with MED numerical semigroups,
  we define Arf systems of generators and minimal Arf generating system for an
  Arf numerical semigroup. These are also unique(see [RGSGGB04]).[133X
  
  [1X8.2-1 IsArf[101X
  
  [29X[2XIsArf[102X( [3XS[103X ) [32X property
  [29X[2XIsArfNumericalSemigroup[102X( [3XS[103X ) [32X property
  
  [33X[0;0Y[3XS[103X  is a numerical semigroup. Returns true if [3XS[103X is an Arf numerical semigroup
  and false otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X IsArfNumericalSemigroup(NumericalSemigroup(3,5,7));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X IsArfNumericalSemigroup(NumericalSemigroup(3,7,11));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMEDNumericalSemigroup(NumericalSemigroup(3,7,11));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 ArfNumericalSemigroupClosure[101X
  
  [29X[2XArfNumericalSemigroupClosure[102X( [3XS[103X ) [32X function
  [29X[2XArfClosure[102X( [3XS[103X ) [32X operation
  
  [33X[0;0Y[3XS[103X is a numerical semigroup. Returns the Arf closure of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11));[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGenerators(last);[127X[104X
    [4X[28X[ 3, 7, 8 ][128X[104X
  [4X[32X[104X
  
  [1X8.2-3 ArfCharactersOfArfNumericalSemigroup[101X
  
  [29X[2XArfCharactersOfArfNumericalSemigroup[102X( [3XS[103X ) [32X function
  [29X[2XMinimalArfGeneratingSystemOfArfNumericalSemigroup[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X  is  an Arf numerical semigroup. Returns the minimal Arf generating system
  of [3XS[103X. The current version of this algorithm is due to G. Zito.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMinimalArfGeneratingSystemOfArfNumericalSemigroup([127X[104X
    [4X[25X>[125X [27XNumericalSemigroup(3,7,8));[127X[104X
    [4X[28X[ 3, 7 ][128X[104X
  [4X[32X[104X
  
  [1X8.2-4 ArfNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XArfNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Y[3Xf[103X  is  an  integer greater than or equal to -1. The output is the set of all
  Arf  numerical  semigroups  with  Frobenius number [3Xf[103X. The current version of
  this algorithm is due to G. Zito.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XArfNumericalSemigroupsWithFrobeniusNumber(10);[127X[104X
    [4X[28X[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,[128X[104X
    [4X[28X  <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,[128X[104X
    [4X[28X  <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ][128X[104X
    [4X[25Xgap>[125X [27XSet(last,MinimalGenerators);[127X[104X
    [4X[28X[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],[128X[104X
    [4X[28X  [ 6, 11, 13, 14, 15, 16 ], [ 7, 9, 11, 12, 13, 15, 17 ],[128X[104X
    [4X[28X  [ 7, 11, 12, 13, 15, 16, 17 ], [ 8, 11, 12, 13, 14, 15, 17, 18 ],[128X[104X
    [4X[28X  [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ], [ 11 .. 21 ] ][128X[104X
  [4X[32X[104X
  
  [1X8.2-5 ArfNumericalSemigroupsWithFrobeniusNumberUpTo[101X
  
  [29X[2XArfNumericalSemigroupsWithFrobeniusNumberUpTo[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Y[3Xf[103X  is  an  integer greater than or equal to -1. The output is the set of all
  Arf  numerical semigroups with Frobenius number less than or equal to [3Xf[103X. The
  current version of this algorithm is due to G. Zito.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(ArfNumericalSemigroupsWithFrobeniusNumberUpTo(10));[127X[104X
    [4X[28X46[128X[104X
  [4X[32X[104X
  
  [1X8.2-6 ArfNumericalSemigroupsWithGenus[101X
  
  [29X[2XArfNumericalSemigroupsWithGenus[102X( [3Xg[103X ) [32X function
  
  [33X[0;0Y[3Xg[103X  is  a  nonnegative  integer.  The  output is the set of all Arf numerical
  semigroups  with equal to [3Xg[103X. The current version of this algorithm is due to
  G. Zito.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(ArfNumericalSemigroupsWithGenus(10));[127X[104X
    [4X[28X21[128X[104X
  [4X[32X[104X
  
  [1X8.2-7 ArfNumericalSemigroupsWithGenusUpTo[101X
  
  [29X[2XArfNumericalSemigroupsWithGenusUpTo[102X( [3Xg[103X ) [32X function
  
  [33X[0;0Y[3Xg[103X  is  a  nonnegative  integer.  The  output is the set of all Arf numerical
  semigroups  with  genus less than or equal to [3Xg[103X. The current version of this
  algorithm is due to G. Zito.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(ArfNumericalSemigroupsWithGenusUpTo(10));[127X[104X
    [4X[28X86[128X[104X
  [4X[32X[104X
  
  [1X8.2-8 ArfNumericalSemigroupsWithGenusAndFrobeniusNumber[101X
  
  [29X[2XArfNumericalSemigroupsWithGenusAndFrobeniusNumber[102X( [3Xg[103X, [3Xf[103X ) [32X function
  
  [33X[0;0Y[3Xf[103X is an integer greater than or equal to -1, and [3Xg[103X is a nonnegative integer.
  The  output  is  the  set  of  all Arf numerical semigroups with genus [3Xg[103X and
  Frobenius number [3Xf[103X. The algorithm is explained in [GSHKR17].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XArfNumericalSemigroupsWithGenusAndFrobeniusNumber(10,13);[127X[104X
    [4X[28X[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,[128X[104X
    [4X[28X  <Numerical semigroup>, <Numerical semigroup> ][128X[104X
    [4X[25Xgap>[125X [27XList(last,MinimalGenerators);[127X[104X
    [4X[28X[ [ 8, 10, 12, 14, 15, 17, 19, 21 ], [ 6, 10, 14, 15, 17, 19 ],[128X[104X
    [4X[28X  [ 5, 12, 14, 16, 18 ], [ 6, 9, 14, 16, 17, 19 ], [ 4, 14, 15, 17 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YSaturated numerical semigroups[133X[101X
  
  [33X[0;0YSaturated  numerical  semigroups with the Arf property are a special kind of
  numerical semigroups with maximal embedding dimension. A numerical semigroup
  [22XS[122X is saturated if the following condition holds: [22Xs, s_1 , ... , s_r[122X in [22XS[122X are
  such  that  [22Xs_i  ≤  s[122X for all [22Xi[122X in [22X{1, ... , r}[122X and [22Xz_1 , ... , z_r[122X in [22XZ[122X are
  such that [22Xz_1 s_1 + ⋯ + z_r s_r≥ 0[122X, then [22Xs + z_1 s_1 + ⋯ + z_r s_r[122X in [22XS[122X.[133X
  
  [33X[0;0YThe  intersection  of two saturated numerical semigroups is again saturated,
  and  thus  we  can  consider  the  saturated closure of a set of nonnegative
  integers with greatest common divisor equal to one (see [RGS09]).[133X
  
  [1X8.3-1 IsSaturated[101X
  
  [29X[2XIsSaturated[102X( [3XS[103X ) [32X property
  [29X[2XIsSaturatedNumericalSemigroup[102X( [3XS[103X ) [32X property
  
  [33X[0;0Y[3XS[103X  is  a  numerical  semigroup.  Returns  true if [3XS[103X is a saturated numerical
  semigroup and false otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSaturatedNumericalSemigroup(NumericalSemigroup(4,6,9,11));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSaturatedNumericalSemigroup(NumericalSemigroup(8, 9, 12, 13, 15, 19 ));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X8.3-2 SaturatedNumericalSemigroupClosure[101X
  
  [29X[2XSaturatedNumericalSemigroupClosure[102X( [3XS[103X ) [32X function
  [29X[2XSaturatedClosure[102X( [3XS[103X ) [32X operation
  
  [33X[0;0Y[3XS[103X is a numerical semigroup. Returns the saturated closure of [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSaturatedNumericalSemigroupClosure(NumericalSemigroup(8, 9, 12, 13, 15));[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGenerators(last);[127X[104X
    [4X[28X[ 8 .. 15 ][128X[104X
  [4X[32X[104X
  
  [1X8.3-3 SaturatedNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XSaturatedNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Y[3Xf[103X  is  an  integer greater than or equal to -1. The output is the set of all
  Saturated numerical semigroups with Frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSaturatedNumericalSemigroupsWithFrobeniusNumber(10);[127X[104X
    [4X[28X[ <Numerical semigroup with 3 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 4 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 6 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 6 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 7 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 8 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 9 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 11 generators> ][128X[104X
    [4X[25Xgap>[125X [27X List(last,MinimalGenerators);[127X[104X
    [4X[28X[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],[128X[104X
    [4X[28X  [ 6, 11, 13, 14, 15, 16 ], [ 7, 11, 12, 13, 15, 16, 17 ],[128X[104X
    [4X[28X  [ 8, 11, 12, 13, 14, 15, 17, 18 ], [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ],[128X[104X
    [4X[28X  [ 11 .. 21 ] ][128X[104X
  [4X[32X[104X
  
