  
  [1X6 [33X[0;0YIrreducible numerical semigroups[133X[101X
  
  
  [1X6.1 [33X[0;0YIrreducible numerical semigroups[133X[101X
  
  [33X[0;0YAn  irreducible  numerical semigroup is a semigroup that cannot be expressed
  as the intersection of numerical semigroups properly containing it.[133X
  
  [33X[0;0YIt  is not difficult to prove that a semigroup is irreducible if and only if
  it  is  maximal  (with respect to set inclusion) in the set of all numerical
  semigroups  having  its same Frobenius number (see [RB03]). Hence, according
  to [FGR87] (respectively [BDF97]), symmetric (respectively pseudo-symmetric)
  numerical  semigroups  are  those  irreducible numerical semigroups with odd
  (respectively even) Frobenius number.[133X
  
  [33X[0;0YIn  [RGSGGJM03]  it  is  shown  that  a  nontrivial  numerical  semigroup is
  irreducible  if  and  only  if  it  has  only  one  special gap. We use this
  characterization.[133X
  
  [33X[0;0YIn this section we show how to construct the set of all numerical semigroups
  with  a  given  Frobenius  number.  In old versions of the package, we first
  constructed  an  irreducible  numerical  semigroup  with the given Frobenius
  number  (as explained in [RGS04]), and then we constructed the rest from it.
  That  is  why we separated both functions. The present version uses a faster
  procedure presented in [BR13].[133X
  
  [33X[0;0YEvery numerical semigroup can be expressed as an intersection of irreducible
  numerical  semigroups.  If  [22XS[122X  can  be  expressed as [22XS=S_1∩ ⋯∩ S_n[122X, with [22XS_i[122X
  irreducible  numerical semigroups, and no factor can be removed, then we say
  that  this  decomposition is minimal. Minimal decompositions can be computed
  by using Algorithm 26 in [RGSGGJM03].[133X
  
  [1X6.1-1 IsIrreducibleNumericalSemigroup[101X
  
  [29X[2XIsIrreducibleNumericalSemigroup[102X( [3Xs[103X ) [32X property
  [29X[2XIsIrreducible[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X  is  a  numerical semigroup. The output is true if [3Xs[103X is irreducible, false
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,9));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,7,9));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 IsSymmetricNumericalSemigroup[101X
  
  [29X[2XIsSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X attribute
  [29X[2XIsSymmetric[102X( [3Xs[103X ) [32X attribute
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup.  The output is true if [3Xs[103X is symmetric, false
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSymmetric(NumericalSemigroup(10,23));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricNumericalSemigroup(NumericalSemigroup(10,11,23));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-3 IsPseudoSymmetric[101X
  
  [29X[2XIsPseudoSymmetric[102X( [3Xs[103X ) [32X property
  [29X[2XIsPseudoSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup. The output is true if [3Xs[103X is pseudo-symmetric,
  false otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(6,7,8,9,11));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(4,6,9));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber[101X
  
  [29X[2XAnIrreducibleNumericalSemigroupWithFrobeniusNumber[102X( [3Xf[103X ) [32X function
  
  [33X[0;0Y[3Xf[103X  is  an  integer greater than or equal to -1. The output is an irreducible
  numerical  semigroup with Frobenius number [3X f[103X. From the way the procedure is
  implemented,  the  resulting  semigroup  has  at  most  four generators (see
  [RGS04]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := AnIrreducibleNumericalSemigroupWithFrobeniusNumber(28);[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XMinimalGenerators(s);[127X[104X
    [4X[28X[ 3, 17, 31 ][128X[104X
    [4X[25Xgap>[125X [27XFrobeniusNumber(s);[127X[104X
    [4X[28X28[128X[104X
  [4X[32X[104X
  
  [1X6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XIrreducibleNumericalSemigroupsWithFrobeniusNumber[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
  irreducible numerical semigroups with Frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(IrreducibleNumericalSemigroupsWithFrobeniusNumber(19));[127X[104X
    [4X[28X20[128X[104X
  [4X[32X[104X
  
  [1X6.1-6 DecomposeIntoIrreducibles[101X
  
  [29X[2XDecomposeIntoIrreducibles[102X( [3Xs[103X ) [32X function
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup. The output is a set of irreducible numerical
  semigroups  containing  it. These elements appear in a minimal decomposition
  of [3Xs[103X as intersection into irreducibles.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDecomposeIntoIrreducibles(NumericalSemigroup(5,6,8));[127X[104X
    [4X[28X[ <Numerical semigroup with 3 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 4 generators> ][128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YComplete intersection numerical semigroups[133X[101X
  
  [33X[0;0YThe  cardinality of a minimal presentation of a numerical semigroup is alwas
  greater  than  or  equal  to  its  embedding  dimension  minus one. Complete
  intersection  numerical  semigroups  are  numerical  semigroups reching this
  bound,  and  they  are  irreducible.  It  can  be  shown that every complete
  intersection  (other that [22XN[122X) is a complete intersection if and only if it is
  the  gluing  of  two complete intersections. When in this gluing, one of the
  copies  is  isomorphic to [22XN[122X, then we obtain a free semigroup in the sense of
  [BC77].  Two  special  kinds  of  free  semigroups are telescopic semigroups
  ([KP95])  and  those associated to an irreducible planar curve ([Zar86]). We
  use  the  algorithms  presented  in  [AGS13] to find the set of all complete
  intersections  (also  free,  telescopic and associated to irreducible planar
  curves) numerical semigroups with given Frobenius number.[133X
  
  [1X6.2-1 AsGluingOfNumericalSemigroups[101X
  
  [29X[2XAsGluingOfNumericalSemigroups[102X( [3Xs[103X ) [32X function
  
  [33X[0;0Y[3Xs[103X  is a numerical semigroup. Returns all partitions [22X{A_1,A_2}[122X of the minimal
  generating  set  of  [3Xs[103X  such  that  [3Xs[103X  is  a  gluing of [22X⟨ A_1⟩[122X and [22X⟨ A_2⟩[122X by
  [22Xgcd(A_1)gcd(A_2)[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroup( 10, 15, 16 );[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XAsGluingOfNumericalSemigroups(s);[127X[104X
    [4X[28X[ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );[127X[104X
    [4X[28X<Numerical semigroup with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XAsGluingOfNumericalSemigroups(s);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [1X6.2-2 IsCompleteIntersection[101X
  
  [29X[2XIsCompleteIntersection[102X( [3Xs[103X ) [32X property
  [29X[2XIsACompleteIntersectionNumericalSemigroup[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  a  complete  intersection,  that  is,  the  cardinality  of  a (any) minimal
  presentation equals its embedding dimension minus one.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroup( 10, 15, 16 );[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsACompleteIntersectionNumericalSemigroup(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );[127X[104X
    [4X[28X<Numerical semigroup with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsACompleteIntersectionNumericalSemigroup(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XCompleteIntersectionNumericalSemigroupsWithFrobeniusNumber[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
  complete intersection numerical semigroups with frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));[127X[104X
    [4X[28X34[128X[104X
  [4X[32X[104X
  
  [1X6.2-4 IsFree[101X
  
  [29X[2XIsFree[102X( [3Xs[103X ) [32X property
  [29X[2XIsFreeNumericalSemigroup[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  free  in  the  sense  of [BC77]: it is either [22XN[122X or the gluing of a copy of [22XN[122X
  with a free numerical semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsFreeNumericalSemigroup(NumericalSemigroup(10,15,16));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XFreeNumericalSemigroupsWithFrobeniusNumber[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
  free numerical semigroups with frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(FreeNumericalSemigroupsWithFrobeniusNumber(57));[127X[104X
    [4X[28X33[128X[104X
  [4X[32X[104X
  
  [1X6.2-6 IsTelescopic[101X
  
  [29X[2XIsTelescopic[102X( [3Xs[103X ) [32X property
  [29X[2XIsTelescopicNumericalSemigroup[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  telescopic  in  the  sense of [KP95]: it is either [22XN[122X or the gluing of [22X⟨ n_e⟩[122X
  and  [22Xs'=⟨  n_1/d,...,  n_e-1/d⟩[122X,  and  [22Xs'[122X  is  again  a telescopic numerical
  semigroup, where [22Xn_1 < ⋯ < n_e[122X are the minimal generators of [3Xs[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsFreeNumericalSemigroup(NumericalSemigroup(4,11,14));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XTelescopicNumericalSemigroupsWithFrobeniusNumber[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
  telescopic numerical semigroups with frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));[127X[104X
    [4X[28X20[128X[104X
  [4X[32X[104X
  
  [1X6.2-8 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity[101X
  
  [29X[2XIsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity[102X( [3Xs[103X ) [32X property
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  associated  to  an  irreducible  planar  curve  singularity ([Zar86]). These
  semigroups are telescopic.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xns := NumericalSemigroup(4,11,14);;[127X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xns := NumericalSemigroup(4,11,19);;[127X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.2-9 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber[101X
  
  [29X[2XNumericalSemigroupsPlanarSingularityWithFrobeniusNumber[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
  numerical  semigroups  associated to irreducible planar curves singularities
  with frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));[127X[104X
    [4X[28X7[128X[104X
  [4X[32X[104X
  
  [1X6.2-10 IsAperySetGammaRectangular[101X
  
  [29X[2XIsAperySetGammaRectangular[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a numerical semigroup.[133X
  
  [33X[0;0YTest  for  the  [22Xγ[122X-rectangularity  of the Apéry Set of a numerical semigroup.
  This test is the implementation of the algorithm given in [DMS14]. Numerical
  Semigroups with this property are free and thus complete intersections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetGammaRectangular(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(4,6,11);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetGammaRectangular(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X                [128X[104X
  [4X[32X[104X
  
  [1X6.2-11 IsAperySetBetaRectangular[101X
  
  [29X[2XIsAperySetBetaRectangular[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a numerical semigroup.[133X
  
  [33X[0;0YTest  for  the  [22Xβ[122X-rectangularity  of the Apéry Set of a numerical semigroup.
  This  test  is  the  implementation  of  the  algorithm  given  in  [DMS14];
  [22Xβ[122X-rectangularity implies [22Xγ[122X-rectangularity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetBetaRectangular(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(4,6,11);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetBetaRectangular(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X                [128X[104X
  [4X[32X[104X
  
  [1X6.2-12 IsAperySetAlphaRectangular[101X
  
  [29X[2XIsAperySetAlphaRectangular[102X( [3XS[103X ) [32X function
  
  [33X[0;0Y[3XS[103X is a numerical semigroup.[133X
  
  [33X[0;0YTest  for  the  [22Xα[122X-rectangularity  of the Apéry Set of a numerical semigroup.
  This  test  is  the  implementation  of  the  algorithm  given  in  [DMS14];
  [22Xα[122X-rectangularity implies [22Xβ[122X-rectangularity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetAlphaRectangular(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(4,6,11);;[127X[104X
    [4X[25Xgap>[125X [27XIsAperySetAlphaRectangular(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YAlmost-symmetric numerical semigroups[133X[101X
  
  [33X[0;0YA  numerical  semigroup  is  almost-symmetric  ([BR97])  if its genus is the
  arithmetic  mean  of  its  Frobenius  number  and  type.  We use a procedure
  presented  in [RGS14] to determine the set of all almost-symmetric numerical
  semigroups  with  given  Frobenius  number.  In  order  to do this, we first
  calculate  the  set of all almost-symmetric numerical semigroups that can be
  constructed from an irreducible numerical semigroup.[133X
  
  [1X6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible[101X
  
  [29X[2XAlmostSymmetricNumericalSemigroupsFromIrreducible[102X( [3Xs[103X ) [32X function
  
  [33X[0;0Y[3Xs[103X  is  an  irreducible  numerical  semigroup.  The  output  is  the  set  of
  almost-symetric  numerical  semigroups  that  can  be  constructed from [3Xs[103X by
  removing some of its generators as explained in [RGS14]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xns := NumericalSemigroup(5,8,9,11);;[127X[104X
    [4X[25Xgap>[125X [27XAlmostSymmetricNumericalSemigroupsFromIrreducible(ns);[127X[104X
    [4X[28X[ <Numerical semigroup with 4 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 5 generators>,[128X[104X
    [4X[28X  <Numerical semigroup with 5 generators> ][128X[104X
    [4X[25Xgap>[125X [27XList(last,MinimalGeneratingSystemOfNumericalSemigroup);[127X[104X
    [4X[28X[ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ][128X[104X
  [4X[32X[104X
  
  [1X6.3-2 IsAlmostSymmetric[101X
  
  [29X[2XIsAlmostSymmetric[102X( [3Xs[103X ) [32X function
  [29X[2XIsAlmostSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X function
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  almost symmetric.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-3 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber[101X
  
  [29X[2XAlmostSymmetricNumericalSemigroupsWithFrobeniusNumber[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
  almost symmetric numerical semigroups with Frobenius number [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12));[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27XLength(IrreducibleNumericalSemigroupsWithFrobeniusNumber(12));[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
