  
  [1X6 [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  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. 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 [33X[0;0YIrreducible numerical semigroups[133X[101X
  
  [33X[0;0YIn this section we provide membership tests to the two families that conform
  the  set  of  irreducible  numerical semigroups. We also give a procedure to
  compute the set of all irreducible numerical semigroups with fixed Frobenius
  number  (or  equivalently  genus, since for irreducible numerical semigroups
  once  the  Frobenius  number  is  fixed,  so  is  the genus). Also we give a
  function  to  compute  the  decomposition  of  a  numerical  semigroup as an
  intersection of irreducible numerical semigroups.[133X
  
  [1X6.1-1 IsIrreducible[101X
  
  [33X[1;0Y[29X[2XIsIrreducible[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsIrreducibleNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical semigroup. The output is [10Xtrue[110X if [3Xs[103X is irreducible, [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YThis   filter   implies   [2XIsAlmostSymmetricNumericalSemigroup[102X   ([14X6.3-3[114X)  and
  [2XIsAcuteNumericalSemigroup[102X ([14X3.1-31[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIrreducible(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 IsSymmetric[101X
  
  [33X[1;0Y[29X[2XIsSymmetric[102X( [3Xs[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X attribute[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup.  The output is [10Xtrue[110X if [3Xs[103X is symmetric, [10Xfalse[110X
  otherwise.[133X
  
  [33X[0;0YThis filter implies [2XIsIrreducibleNumericalSemigroup[102X ([14X6.1-1[114X).[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
  
  [33X[1;0Y[29X[2XIsPseudoSymmetric[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPseudoSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup. The output is [10Xtrue[110X if [3Xs[103X is pseudo-symmetric,
  [10Xfalse[110X otherwise.[133X
  
  [33X[0;0YThis filter implies [2XIsIrreducibleNumericalSemigroup[102X ([14X6.1-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPseudoSymmetric(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
  
  [33X[1;0Y[29X[2XAnIrreducibleNumericalSemigroupWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an  integer.  When  [22Xf=0[122X or [22Xfle -2[122X, the output is [10Xfail[110X. Otherwise, the
  output  is  an irreducible numerical semigroup with Frobenius number [3Xf[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
  
  [33X[1;0Y[29X[2XIrreducibleNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an  integer.  The  output  is  the  set  of all irreducible numerical
  semigroups with Frobenius number [3Xf[103X. The algorithm is inspired in [BR13].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(IrreducibleNumericalSemigroupsWithFrobeniusNumber(19));[127X[104X
    [4X[28X20[128X[104X
  [4X[32X[104X
  
  [1X6.1-6 IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity[101X
  
  [33X[1;0Y[29X[2XIrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity[102X( [3Xf[103X, [3Xm[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  and  [3Xm[103X  are  integers. The output is the set of all irreducible numerical
  semigroups  with  Frobenius  number [3Xf[103X and multiplicity [3Xm[103X. The implementation
  appears in [ttt19].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity(31,11));[127X[104X
    [4X[28X16[128X[104X
  [4X[32X[104X
  
  [1X6.1-7 DecomposeIntoIrreducibles[101X
  
  [33X[1;0Y[29X[2XDecomposeIntoIrreducibles[102X( [3Xs[103X ) [32X function[133X
  
  [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 always
  greater  than  or  equal  to  its  embedding  dimension  minus one. Complete
  intersection  numerical  semigroups  are  numerical semigroups reaching 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
  
  [33X[1;0Y[29X[2XAsGluingOfNumericalSemigroups[102X( [3Xs[103X ) [32X function[133X
  
  [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
  
  [33X[1;0Y[29X[2XIsCompleteIntersection[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsACompleteIntersectionNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [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
  
  [33X[0;0YThis    filter    implies    [2XIsSymmetricNumericalSemigroup[102X    ([14X6.1-2[114X)    and
  [2XIsCyclotomicNumericalSemigroup[102X ([14X10.1-8[114X).[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 [27XIsCompleteIntersection(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
  
  [33X[1;0Y[29X[2XCompleteIntersectionNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an  integer.  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
  
  [33X[1;0Y[29X[2XIsFree[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsFreeNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [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
  
  [33X[0;0YThis filter implies [2XIsACompleteIntersectionNumericalSemigroup[102X ([14X6.2-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsFree(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
  
  [33X[1;0Y[29X[2XFreeNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X is an integer. 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
  
  [33X[1;0Y[29X[2XIsTelescopic[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsTelescopicNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [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
  
  [33X[0;0YThis filter implies [2XIsAperySetBetaRectangular[102X ([14X6.2-12[114X) and [2XIsFree[102X ([14X6.2-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsTelescopic(NumericalSemigroup(4,11,14));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsFree(NumericalSemigroup(4,11,14));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber[101X
  
  [33X[1;0Y[29X[2XTelescopicNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an  integer.  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 IsUniversallyFree[101X
  
  [33X[1;0Y[29X[2XIsUniversallyFree[102X( [3Xs[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsUniversallyFreeNumericalSemigroup[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is true if the numerical semigroup is
  free for all the arrangements of its minimal generators.[133X
  
  [33X[0;0YThis filter implies [2XIsTelescopic[102X ([14X6.2-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(10,15,18);;[127X[104X
    [4X[25Xgap>[125X [27XIsUniversallyFree(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(4,6,9);;[127X[104X
    [4X[25Xgap>[125X [27XIsFree(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsUniversallyFree(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-9 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity[101X
  
  [33X[1;0Y[29X[2XIsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity[102X( [3Xs[103X ) [32X property[133X
  
  [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
  
  [33X[0;0YThis     filter     implies    [2XIsAperySetAlphaRectangular[102X    ([14X6.2-13[114X)    and
  [2XIsTelescopicNumericalSemigroup[102X ([14X6.2-6[114X).[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
  
  [1X6.2-10 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupsPlanarSingularityWithFrobeniusNumber[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an  integer.  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
  
  [1X6.2-11 IsAperySetGammaRectangular[101X
  
  [33X[1;0Y[29X[2XIsAperySetGammaRectangular[102X( [3XS[103X ) [32X function[133X
  
  [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
  
  [33X[0;0YThis filter implies [2XIsFreeNumericalSemigroup[102X ([14X6.2-4[114X).[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
  
  [1X6.2-12 IsAperySetBetaRectangular[101X
  
  [33X[1;0Y[29X[2XIsAperySetBetaRectangular[102X( [3XS[103X ) [32X function[133X
  
  [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
  
  [33X[0;0YThis filter implies [2XIsAperySetGammaRectangular[102X ([14X6.2-11[114X).[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
  
  [1X6.2-13 IsAperySetAlphaRectangular[101X
  
  [33X[1;0Y[29X[2XIsAperySetAlphaRectangular[102X( [3XS[103X ) [32X function[133X
  
  [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
  
  [33X[0;0YThis filter implies [2XIsAperySetBetaRectangular[102X ([14X6.2-12[114X).[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  ([BF97])  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
  
  [33X[1;0Y[29X[2XAlmostSymmetricNumericalSemigroupsFromIrreducible[102X( [3Xs[103X ) [32X function[133X
  
  [33X[0;0Y[3Xs[103X  is  an  irreducible  numerical  semigroup.  The  output  is  the  set  of
  almost-symmetric  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 AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType[101X
  
  [33X[1;0Y[29X[2XAlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType[102X( [3Xs[103X, [3Xt[103X ) [32X function[133X
  
  [33X[0;0Y[3Xs[103X  is  an  irreducible  numerical semigroup and [3Xt[103X is a positive integer. The
  output  is the set of almost-symmetric numerical semigroups with type [3Xt[103X that
  can  be  constructed from [3Xs[103X by removing some of its generators (as explained
  in [BOR18]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xns := NumericalSemigroup(5,8,9,11);;[127X[104X
    [4X[25Xgap>[125X [27XAlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType(ns,4);[127X[104X
    [4X[28X[ <Numerical semigroup with 5 generators>, [128X[104X
    [4X[28X  <Numerical semigroup with 5 generators> ][128X[104X
    [4X[25Xgap>[125X [27XList(last,MinimalGenerators);[127X[104X
    [4X[28X[ [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ][128X[104X
  [4X[32X[104X
  
  [1X6.3-3 IsAlmostSymmetric[101X
  
  [33X[1;0Y[29X[2XIsAlmostSymmetric[102X( [3Xs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XIsAlmostSymmetricNumericalSemigroup[102X( [3Xs[103X ) [32X function[133X
  
  [33X[0;0Y[3Xs[103X is a numerical semigroup. The output is [10Xtrue[110X if the numerical semigroup is
  almost symmetric.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetric(NumericalSemigroup(5,8,11,14,17));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-4 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber[101X
  
  [33X[1;0Y[29X[2XAlmostSymmetricNumericalSemigroupsWithFrobeniusNumber[102X( [3Xf[103X[, [3Xts[103X] ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X is an integer, and so is [3Xts[103X. The output is the set of all almost symmetric
  numerical semigroups with Frobenius number [3Xf[103X, and type greater than or equal
  to [3Xts[103X. If [3Xts[103X is not specified, then it is considered to be equal to one, and
  so  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[25Xgap>[125X [27XList(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12,4),Type);[127X[104X
    [4X[28X[ 12, 10, 8, 8, 6, 6, 6, 6, 4, 4, 4, 4, 4 ][128X[104X
  [4X[32X[104X
  
  [1X6.3-5 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType[101X
  
  [33X[1;0Y[29X[2XAlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType[102X( [3Xf[103X, [3Xt[103X ) [32X function[133X
  
  [33X[0;0Y[3Xf[103X  is  an integer and so is [3Xt[103X. The output is the set of all almost symmetric
  numerical semigroups with Frobenius number [3Xf[103X and type [3Xt[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType(12,4)); [127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YSeveral approaches generalizing the concept of symmetry[133X[101X
  
  [33X[0;0YLet  [22XS[122X  be  a numerical semigroup and let [22XR[122X be its semigroup ring [22XK[[S]][122X. We
  say that [22XS[122X has the generalized Gorenstein property if its semigroup ring has
  this  property.  For  the  definition  and  characterization  of generalized
  Gorenstein rings please see [tttt17].[133X
  
  [33X[0;0YA  numerical  semigroup is said to be nearly Gorenstein if its maximal ideal
  is  contained  in  its trace ideal [HHS19]. Every almost symmetric numerical
  semigroup is nearly Gorenstein.[133X
  
  [33X[0;0YA  numerical  semigroup  [22XS[122X  with  canonical  ideal [22XK[122X is a generalized almost
  symmetric   numerical   semigroup   if   either   [22X2K=K[122X  (symmetric)  or  [22X2K∖
  K={F(S)-x_1,dots,  F(S)-x_r,F(S)}[122X  for  some  [22Xx_1,dots,x_r  ∈ M∖ 2M[122X (minimal
  generators)  and  [22Xx_i-x_jnot\in (S-M)∖ S[122X (not pseudo-Frobenius numbers), see
  [DS21].  As  expected,  every  almost  symmetric  numerical  semigroup  is a
  generalized almost symmetric numerical semigroup.[133X
  
  [1X6.4-1 IsGeneralizedGorenstein[101X
  
  [33X[1;0Y[29X[2XIsGeneralizedGorenstein[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X  is a numerical semigroup. The output is [10Xtrue[110X if the semigroup ring [22XK[[S]][122X
  is  generalized  Gorenstein  using  the  characterization  by Goto-Kumashiro
  [MK17].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,7,8);;[127X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetric(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsGeneralizedGorenstein(s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.4-2 IsNearlyGorenstein[101X
  
  [33X[1;0Y[29X[2XIsNearlyGorenstein[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical semigroup. The output is [10Xtrue[110X if the semigroup is nearly
  Gorenstein, and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(10,11,12,25);;[127X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetric(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNearlyGorenstein(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,7,8);;[127X[104X
    [4X[25Xgap>[125X [27XIsNearlyGorenstein(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-3 NearlyGorensteinVectors[101X
  
  [33X[1;0Y[29X[2XNearlyGorensteinVectors[102X( [3Xs[103X ) [32X operation[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup.  The  output is a lists of lists (making the
  cartesian product of them yields all possible NG-vectors). If [22Xn_i[122X is the ith
  generator   of   [3Xs[103X,  in  the  ith  position  of  the  list  it  returns  all
  pseudo-Frobenius  numbers  [22Xf[122X  of  [3Xs[103X  such  that [22Xn_i+f-f'[122X is in [3Xs[103X for all [22Xf[122X a
  pseudo-Frobenius number of [3Xs[103X, [MS21].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(10,11,12,25);;[127X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetric(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNearlyGorenstein(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,7,8);;[127X[104X
    [4X[25Xgap>[125X [27XIsNearlyGorenstein(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-4 IsGeneralizedAlmostSymmetric[101X
  
  [33X[1;0Y[29X[2XIsGeneralizedAlmostSymmetric[102X( [3Xs[103X ) [32X property[133X
  
  [33X[0;0Y[3Xs[103X  is  a  numerical  semigroup. Determines whether or not [3Xs[103X is a generalized
  almost symmetric numerical semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(9, 24, 39, 43, 77);;[127X[104X
    [4X[25Xgap>[125X [27XIsGeneralizedAlmostSymmetric(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAlmostSymmetric(s);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
