chap4.txt

  
  4 Presentations of Numerical Semigroups
  
  In  this  chapter  we  explain  how  to  compute a minimal presentation of a
  numerical semigroup. There are three functions involved in this process.
  
  
  4.1 Presentations of Numerical Semigroups
  
  4.1-1 MinimalPresentationOfNumericalSemigroup
  
  MinimalPresentationOfNumericalSemigroup( S )  function
  MinimalPresentation( S )  operation
  
  S is a numerical semigroup. The output is a list of lists with two elements.
  Each  list  of  two  elements  represents  a  relation  between  the minimal
  generators  of the numerical semigroup. If { {x_1,y_1},...,{x_k,y_k}} is the
  output  and  {m_1,...,m_n}  is  the  minimal  system  of  generators  of the
  numerical  semigroup,  then  {x_i,y_i}={{a_i_1,...,a_i_n},{b_i_1,...,b_i_n}}
  and a_i_1m_1+⋯+a_i_nm_n= b_i_1m_1+ ⋯ +b_i_nm_n.
  
  Any  other  relation  among  the  minimal generators of the semigroup can be
  deduced from the ones given in the output.
  
  The algorithm implemented is described in [Ros96a] (see also [RGS99b]).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);
    <Numerical semigroup with 3 generators>
    gap> MinimalPresentationOfNumericalSemigroup(s);
    [ [ [ 0, 2, 0 ], [ 1, 0, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ],
    [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
  
  
  The first element in the list means that 1× 3+1× 7=2× 5, and the others have
  similar meanings.
  
  4.1-2 GraphAssociatedToElementInNumericalSemigroup
  
  GraphAssociatedToElementInNumericalSemigroup( n, S )  function
  
  S is a numerical semigroup and n is an element in S.
  
  The  output  is a pair. If {m_1,...,m_n} is the set of minimal generators of
  S,  then  the first component is the set of vertices of the graph associated
  to  n  in S, that is, the set { m_i | n-m_i∈ S}, and the second component is
  the set of edges of this graph, that is, { {m_i,m_j} | n-(m_i+m_j)∈ S}.
  
  This  function  is  used  to compute a minimal presentation of the numerical
  semigroup S, as explained in [Ros96a].
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> GraphAssociatedToElementInNumericalSemigroup(10,s);
    [ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
  
  
  4.1-3 BettiElementsOfNumericalSemigroup
  
  BettiElementsOfNumericalSemigroup( S )  function
  BettiElements( S )  operation
  
  S is a numerical semigroup.
  
  The  output  is  the  set  of  elements  in  S  whose  associated  graph  is
  nonconnected [GSO10].
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> BettiElementsOfNumericalSemigroup(s);
    [ 10, 12, 14 ]
  
  
  4.1-4 PrimitiveElementsOfNumericalSemigroup
  
  PrimitiveElementsOfNumericalSemigroup( S )  function
  
  S is a numerical semigroup.
  
  The  output  is  the set of elements s in S such that there exists a minimal
  solution  to  msg⋅  x-msg⋅ y = 0, such that x,y are factorizations of s, and
  msg is the minimal generating system of S. Betti elements are primitive, but
  not the way around in general.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> PrimitiveElementsOfNumericalSemigroup(s);
    [ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
  
  
  4.1-5 ShadedSetOfElementInNumericalSemigroup
  
  ShadedSetOfElementInNumericalSemigroup( n, S )  function
  
  S is a numerical semigroup and n is an element in S.
  
  The output is a simplicial complex C. If {m_1,...,m_n} is the set of minimal
  generators of S, then L ∈ C if n-∑_i∈ L m_i∈ S ([SW86]).
  
  This function is a generalization of the graph associated to n.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> ShadedSetOfElementInNumericalSemigroup(10,s);
    [ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
  
  
  
  4.2 Uniquely Presented Numerical Semigroups
  
  A  numerical  semigroup  S  is  uniquely  presented  if  for any two minimal
  presentations  σ  and  τ and any (a,b)∈ σ, either (a,b)∈ τ or (b,a)∈ τ, that
  is, there is essentially a unique minimal presentation (up to arrangement of
  the components of the pairs in it).
  
  4.2-1 IsUniquelyPresented
  
  IsUniquelyPresented( S )  property
  IsUniquelyPresentedNumericalSemigroup( S )  property
  
  S is a numerical semigroup.
  
  The  output is true if S has uniquely presented. The implementation is based
  on (see [GSO10]).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> IsUniquelyPresentedNumericalSemigroup(s);
    true
  
  
  4.2-2 IsGeneric
  
  IsGeneric( S )  property
  IsGenericNumericalSemigroup( S )  property
  
  S is a numerical semigroup.
  
  The  output  is  true  if  S  has  a generic presentation, that is, in every
  minimal  relation  all  generators  occur.  These  semigroups  are  uniquely
  presented (see [BGSG11]).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> IsGenericNumericalSemigroup(s);
    true