chap6.txt

  
  6 Irreducible numerical semigroups
  
  
  6.1 Irreducible numerical semigroups
  
  An  irreducible  numerical semigroup is a semigroup that cannot be expressed
  as the intersection of numerical semigroups properly containing it.
  
  It  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.
  
  In  [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.
  
  In 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].
  
  Every numerical semigroup can be expressed as an intersection of irreducible
  numerical  semigroups.  If  S  can  be  expressed as S=S_1∩ ⋯∩ S_n, with S_i
  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].
  
  6.1-1 IsIrreducibleNumericalSemigroup
  
  IsIrreducibleNumericalSemigroup( s )  property
  IsIrreducible( s )  property
  
  s  is  a  numerical semigroup. The output is true if s is irreducible, false
  otherwise.
  
    Example  
    gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,9));
    true
    gap> IsIrreducibleNumericalSemigroup(NumericalSemigroup(4,6,7,9));
    false
  
  
  6.1-2 IsSymmetricNumericalSemigroup
  
  IsSymmetricNumericalSemigroup( s )  attribute
  IsSymmetric( s )  attribute
  
  s  is  a  numerical  semigroup.  The output is true if s is symmetric, false
  otherwise.
  
    Example  
    gap> IsSymmetric(NumericalSemigroup(10,23));
    true
    gap> IsSymmetricNumericalSemigroup(NumericalSemigroup(10,11,23));
    false
  
  
  6.1-3 IsPseudoSymmetric
  
  IsPseudoSymmetric( s )  property
  IsPseudoSymmetricNumericalSemigroup( s )  property
  
  s  is  a  numerical  semigroup. The output is true if s is pseudo-symmetric,
  false otherwise.
  
    Example  
    gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(6,7,8,9,11));
    true
    gap> IsPseudoSymmetricNumericalSemigroup(NumericalSemigroup(4,6,9));
    false
  
  
  6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber
  
  AnIrreducibleNumericalSemigroupWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is an irreducible
  numerical  semigroup with Frobenius number  f. From the way the procedure is
  implemented,  the  resulting  semigroup  has  at  most  four generators (see
  [RGS04]).
  
    Example  
    gap> s := AnIrreducibleNumericalSemigroupWithFrobeniusNumber(28);
    <Numerical semigroup with 3 generators>
    gap> MinimalGenerators(s);
    [ 3, 17, 31 ]
    gap> FrobeniusNumber(s);
    28
  
  
  6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber
  
  IrreducibleNumericalSemigroupsWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is the set of all
  irreducible numerical semigroups with Frobenius number f.
  
    Example  
    gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(19));
    20
  
  
  6.1-6 DecomposeIntoIrreducibles
  
  DecomposeIntoIrreducibles( s )  function
  
  s  is  a  numerical  semigroup. The output is a set of irreducible numerical
  semigroups  containing  it. These elements appear in a minimal decomposition
  of s as intersection into irreducibles.
  
    Example  
    gap> DecomposeIntoIrreducibles(NumericalSemigroup(5,6,8));
    [ <Numerical semigroup with 3 generators>,
      <Numerical semigroup with 4 generators> ]
  
  
  
  6.2 Complete intersection numerical semigroups
  
  The  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 N) 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 N, 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.
  
  6.2-1 AsGluingOfNumericalSemigroups
  
  AsGluingOfNumericalSemigroups( s )  function
  
  s  is a numerical semigroup. Returns all partitions {A_1,A_2} of the minimal
  generating  set  of  s  such  that  s  is  a  gluing of ⟨ A_1⟩ and ⟨ A_2⟩ by
  gcd(A_1)gcd(A_2)
  
    Example  
    gap> s := NumericalSemigroup( 10, 15, 16 );
    <Numerical semigroup with 3 generators>
    gap> AsGluingOfNumericalSemigroups(s);
    [ [ [ 10, 15 ], [ 16 ] ], [ [ 10, 16 ], [ 15 ] ] ]
    gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
    <Numerical semigroup with 8 generators>
    gap> AsGluingOfNumericalSemigroups(s);
    [  ]
  
  
  6.2-2 IsCompleteIntersection
  
  IsCompleteIntersection( s )  property
  IsACompleteIntersectionNumericalSemigroup( s )  property
  
  s 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.
  
    Example  
    gap> s := NumericalSemigroup( 10, 15, 16 );
    <Numerical semigroup with 3 generators>
    gap> IsACompleteIntersectionNumericalSemigroup(s);
    true
    gap> s := NumericalSemigroup( 18, 24, 34, 46, 51, 61, 74, 8 );
    <Numerical semigroup with 8 generators>
    gap> IsACompleteIntersectionNumericalSemigroup(s);
    false
  
  
  6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber
  
  CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is the set of all
  complete intersection numerical semigroups with frobenius number f.
  
    Example  
    gap> Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(57));
    34
  
  
  6.2-4 IsFree
  
  IsFree( s )  property
  IsFreeNumericalSemigroup( s )  property
  
  s is a numerical semigroup. The output is true if the numerical semigroup is
  free  in  the  sense  of [BC77]: it is either N or the gluing of a copy of N
  with a free numerical semigroup.
  
    Example  
    gap> IsFreeNumericalSemigroup(NumericalSemigroup(10,15,16));
    true
    gap> IsFreeNumericalSemigroup(NumericalSemigroup(3,5,7));
    false
  
  
  6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber
  
  FreeNumericalSemigroupsWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is the set of all
  free numerical semigroups with frobenius number f.
  
    Example  
    gap> Length(FreeNumericalSemigroupsWithFrobeniusNumber(57));
    33
  
  
  6.2-6 IsTelescopic
  
  IsTelescopic( s )  property
  IsTelescopicNumericalSemigroup( s )  property
  
  s is a numerical semigroup. The output is true if the numerical semigroup is
  telescopic  in  the  sense of [KP95]: it is either N or the gluing of ⟨ n_e⟩
  and  s'=⟨  n_1/d,...,  n_e-1/d⟩,  and  s'  is  again  a telescopic numerical
  semigroup, where n_1 < ⋯ < n_e are the minimal generators of s.
  
    Example  
    gap> IsTelescopicNumericalSemigroup(NumericalSemigroup(4,11,14));
    false
    gap> IsFreeNumericalSemigroup(NumericalSemigroup(4,11,14));
    true
  
  
  6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber
  
  TelescopicNumericalSemigroupsWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is the set of all
  telescopic numerical semigroups with frobenius number f.
  
    Example  
    gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(57));
    20
  
  
  6.2-8 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity
  
  IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity( s )  property
  
  s 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.
  
    Example  
    gap> ns := NumericalSemigroup(4,11,14);;
    gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
    false
    gap> ns := NumericalSemigroup(4,11,19);;
    gap> IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(ns);
    true
  
  
  6.2-9 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber
  
  NumericalSemigroupsPlanarSingularityWithFrobeniusNumber( f )  function
  
  f  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 f.
  
    Example  
    gap> Length(NumericalSemigroupsPlanarSingularityWithFrobeniusNumber(57));
    7
  
  
  6.2-10 IsAperySetGammaRectangular
  
  IsAperySetGammaRectangular( S )  function
  
  S is a numerical semigroup.
  
  Test  for  the  γ-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.
  
    Example  
    gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
    gap> IsAperySetGammaRectangular(s);
    false
    gap> s:=NumericalSemigroup(4,6,11);;
    gap> IsAperySetGammaRectangular(s);
    true
                    
  
  
  6.2-11 IsAperySetBetaRectangular
  
  IsAperySetBetaRectangular( S )  function
  
  S is a numerical semigroup.
  
  Test  for  the  β-rectangularity  of the Apéry Set of a numerical semigroup.
  This  test  is  the  implementation  of  the  algorithm  given  in  [DMS14];
  β-rectangularity implies γ-rectangularity.
  
    Example  
    gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
    gap> IsAperySetBetaRectangular(s);
    false
    gap> s:=NumericalSemigroup(4,6,11);;
    gap> IsAperySetBetaRectangular(s);
    true
                    
  
  
  6.2-12 IsAperySetAlphaRectangular
  
  IsAperySetAlphaRectangular( S )  function
  
  S is a numerical semigroup.
  
  Test  for  the  α-rectangularity  of the Apéry Set of a numerical semigroup.
  This  test  is  the  implementation  of  the  algorithm  given  in  [DMS14];
  α-rectangularity implies β-rectangularity.
  
    Example  
    gap> s:=NumericalSemigroup(30, 35, 42, 47, 148, 153, 157, 169, 181, 193);;
    gap> IsAperySetAlphaRectangular(s);
    false
    gap> s:=NumericalSemigroup(4,6,11);;
    gap> IsAperySetAlphaRectangular(s);
    true
  
  
  
  6.3 Almost-symmetric numerical semigroups
  
  A  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.
  
  6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
  
  AlmostSymmetricNumericalSemigroupsFromIrreducible( s )  function
  
  s  is  an  irreducible  numerical  semigroup.  The  output  is  the  set  of
  almost-symetric  numerical  semigroups  that  can  be  constructed from s by
  removing some of its generators as explained in [RGS14]).
  
    Example  
    gap> ns := NumericalSemigroup(5,8,9,11);;
    gap> AlmostSymmetricNumericalSemigroupsFromIrreducible(ns);
    [ <Numerical semigroup with 4 generators>,
      <Numerical semigroup with 5 generators>,
      <Numerical semigroup with 5 generators> ]
    gap> List(last,MinimalGeneratingSystemOfNumericalSemigroup);
    [ [ 5, 8, 9, 11 ], [ 5, 8, 11, 14, 17 ], [ 5, 9, 11, 13, 17 ] ]
  
  
  6.3-2 IsAlmostSymmetric
  
  IsAlmostSymmetric( s )  function
  IsAlmostSymmetricNumericalSemigroup( s )  function
  
  s is a numerical semigroup. The output is true if the numerical semigroup is
  almost symmetric.
  
    Example  
    gap> IsAlmostSymmetricNumericalSemigroup(NumericalSemigroup(5,8,11,14,17));
    true
  
  
  6.3-3 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
  
  AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber( f )  function
  
  f  is  an  integer greater than or equal to -1. The output is the set of all
  almost symmetric numerical semigroups with Frobenius number f.
  
    Example  
    gap> Length(AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber(12));
    15
    gap> Length(IrreducibleNumericalSemigroupsWithFrobeniusNumber(12));
    2