3 Basic operations with numerical semigroups
  
  
  3.1 The definitions
  
  3.1-1 MultiplicityOfNumericalSemigroup
  
  > MultiplicityOfNumericalSemigroup( NS ) __________________________attribute
  
  NS  is  a  numerical semigroup. Returns the multiplicity of NS, which is the
  smallest positive integer belonging to NS.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 7,53);
    <Modular numerical semigroup satisfying 7x mod 53 <= x >
    gap> MultiplicityOfNumericalSemigroup(S);
    8
  ------------------------------------------------------------------
  
  3.1-2 GeneratorsOfNumericalSemigroup
  
  > GeneratorsOfNumericalSemigroup( S ) ______________________________function
  > GeneratorsOfNumericalSemigroupNC( S ) ____________________________function
  > MinimalGeneratingSystemOfNumericalSemigroup( S ) ________________attribute
  
  S  is a numerical semigroup. GeneratorsOfNumericalSemigroup returns a set of
  generators  of S, which may not be minimal. GeneratorsOfNumericalSemigroupNC
  returns  the  set  of generators recorded in S!.generators, which may not be
  minimal. MinimalGeneratingSystemOfNumericalSemigroup returns the minimal set
  of generators of S.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 5,53);
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
    gap> GeneratorsOfNumericalSemigroup(S);
    [ 11, 12, 13, 32, 53 ]
    gap> S := NumericalSemigroup(3, 5, 53);
    <Numerical semigroup with 3 generators>
    gap> GeneratorsOfNumericalSemigroup(S);
    [ 3, 5, 53 ]
    gap> MinimalGeneratingSystemOfNumericalSemigroup(S);
    [ 3, 5 ]
  ------------------------------------------------------------------
  
  3.1-3 SmallElementsOfNumericalSemigroup
  
  > SmallElementsOfNumericalSemigroup( NS ) _________________________attribute
  
  NS is a numerical semigroup. It returns the list of small elements of NS. Of
  course,  the  time  consumed  to  return  a result may depend on the way the
  semigroup is given.
  
  ---------------------------  Example  ----------------------------
    gap> SmallElementsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
    [ 0, 3, 5 ]
  ------------------------------------------------------------------
  
  3.1-4 FirstElementsOfNumericalSemigroup
  
  > FirstElementsOfNumericalSemigroup( n, NS ) _______________________function
  
  NS  is  a numerical semigroup. It returns the list with the first n elements
  of NS.
  
  ---------------------------  Example  ----------------------------
    gap> FirstElementsOfNumericalSemigroup(2,NumericalSemigroup(3,5,7));
    [ 0, 3 ]
    gap> FirstElementsOfNumericalSemigroup(10,NumericalSemigroup(3,5,7));
    [ 0, 3, 5, 6, 7, 8, 9, 10, 11, 12 ]
  ------------------------------------------------------------------
  
  3.1-5 AperyListOfNumericalSemigroupWRTElement
  
  > AperyListOfNumericalSemigroupWRTElement( S, m ) _________________operation
  
  S  is  a  numerical semigroup and m is a positive element of S. Computes the
  Apéry  list  of S wrt m. It contains for every iin {0,...,m-1}, in the i+1th
  position, the smallest element in the semigroup congruent with i modulo m.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 5,53);
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
    gap> AperyListOfNumericalSemigroupWRTElement(S,12);
    [ 0, 13, 26, 39, 52, 53, 54, 43, 32, 33, 22, 11 ]
  ------------------------------------------------------------------
  
  3.1-6 DrawAperyListOfNumericalSemigroup
  
  > DrawAperyListOfNumericalSemigroup( ap ) __________________________function
  
  ap is the Apéry list of a numerical semigroup. This function draws the graph
  (ap,  E)  where  the  edge  u  ->  v is in E iff v - u is in ap. To use this
  function,  Graphviz  (http://www.graphviz.org)  should be installed and also
  Evince         (http://www.gnome.org/projects/evince/)         or        ggv
  (http://directory.fsf.org/ggv.html).
  
  3.1-7 AperyListOfNumericalSemigroupAsGraph
  
  > AperyListOfNumericalSemigroupAsGraph( ap ) _______________________function
  
  ap  is  the  Apéry  list of a numerical semigroup. This function returns the
  adjacency  list of the graph (ap, E) where the edge u -> v is in E iff v - u
  is in ap. The 0 is ignored.
  
  ---------------------------  Example  ----------------------------
    gap> s:=NumericalSemigroup(3,7);
    <Numerical semigroup with 2 generators>
    gap> AperyListOfNumericalSemigroupWRTElement(s,10);
    [ 0, 21, 12, 3, 14, 15, 6, 7, 18, 9 ]
    gap> AperyListOfNumericalSemigroupAsGraph(last);
    [ ,, [ 3, 6, 9, 12, 15, 18, 21 ],,, [ 6, 9, 12, 15, 18, 21 ],
    [ 7, 14, 21 ],, [ 9, 12, 15, 18, 21 ],,, [ 12, 15, 18, 21 ],,
    [ 14, 21 ], [ 15, 18, 21 ],,, [ 18, 21 ],,, [ 21 ] ]
  ------------------------------------------------------------------
  
  
  3.2 Frobenius Number
  
  The  largest nonnegative integer not belonging to a numerical semigroup S is
  the  Frobenius  number  of  S. If S is the set of nonnegative integers, then
  clearly its Frobenius number is -1, otherwise its Frobenius number coincides
  with  the  maximum of the gaps (or fundamental gaps) of S. An integer z is a
  pseudo-Frobenius number of S if z+S\{0}subseteq S.
  
  3.2-1 FrobeniusNumberOfNumericalSemigroup
  
  > FrobeniusNumberOfNumericalSemigroup( NS ) _______________________attribute
  
  NS  is  a  numerical  semigroup.  It  returns the Frobenius number of NS. Of
  course,  the  time  consumed  to  return  a result may depend on the way the
  semigroup is given or on the knowledge already produced on the semigroup.
  
  ---------------------------  Example  ----------------------------
    gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));
    4
  ------------------------------------------------------------------
  
  3.2-2 FrobeniusNumber
  
  > FrobeniusNumber( NS ) ___________________________________________attribute
  
  This is just a synonym of FrobeniusNumberOfNumericalSemigroup (3.2-1).
  
  3.2-3 PseudoFrobeniusOfNumericalSemigroup
  
  > PseudoFrobeniusOfNumericalSemigroup( S ) ________________________attribute
  
  S is a numerical semigroup. It returns set of pseudo-Frobenius numbers of S.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 5,53);
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
    gap> PseudoFrobeniusOfNumericalSemigroup(S);
    [ 21, 40, 41, 42 ]
  ------------------------------------------------------------------
  
  
  3.3 Gaps
  
  A  gap  of a numerical semigroup S is a nonnegative integer not belonging to
  S. The fundamental gaps of S are those gaps that are maximal with respect to
  the  partial order induced by division in N. The special gaps of a numerical
  semigroup  S,  are those fundamental gaps such that if they are added to the
  given  numerical  semigroup,  then  the  resulting  set is again a numerical
  semigroup.
  
  3.3-1 GapsOfNumericalSemigroup
  
  > GapsOfNumericalSemigroup( NS ) __________________________________attribute
  
  NS is a numerical semigroup. It returns the set of gaps of NS.
  
  ---------------------------  Example  ----------------------------
    gap> GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));
    [ 1, 2, 4 ]
  ------------------------------------------------------------------
  
  3.3-2 FundamentalGapsOfNumericalSemigroup
  
  > FundamentalGapsOfNumericalSemigroup( S ) ________________________attribute
  
  S is a numerical semigroup. It returns the set of fundamental gaps of S.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 5,53);
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
    gap> FundamentalGapsOfNumericalSemigroup(S);
    [ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ]
    gap> GapsOfNumericalSemigroup(S);
    [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29,
      30, 31, 40, 41, 42 ]
  ------------------------------------------------------------------
  
  3.3-3 SpecialGapsOfNumericalSemigroup
  
  > SpecialGapsOfNumericalSemigroup( S ) ____________________________attribute
  
  S is a numerical semigroup. It returns the special gaps of S.
  
  ---------------------------  Example  ----------------------------
    gap> S := NumericalSemigroup("modular", 5,53);
    <Modular numerical semigroup satisfying 5x mod 53 <= x >
    gap> SpecialGapsOfNumericalSemigroup(S);
    [ 40, 41, 42 ]
  ------------------------------------------------------------------