GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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  11 Affine semigroups
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  11.1 Defining affine semigroups
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  An  affine  semigroup  S is a finitely generated cancellative monoid that is
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  reduced  (no  units other than 0) and is torsion-free (a s= b s implies a=b,
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  with a,b∈ N and s∈ S). Up to isomorphisms any affine semigroup can be viewed
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  as  a  finitely generated submonoid of N^k for some positive integer k. Thus
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  affine  semigroups are a natural generalization of numerical semigroups. The
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  most  common  way  to  give  an affine semigroup is by any of its systems of
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  generators.  As  for  numerical  semigroups,  any  affine semigroup admits a
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  unique  minimal  system  of  generators.  A  system  of  generators  can  be
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  represented  as  a  list  of lists of nonnegative integers; all lists in the
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  list  having the same length (a matrix actually). If G is a subgroup of Z^k,
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  then  S=G∩  N^k  is  an  affine  semigroup (these semigroups are called full
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  affine  semigroups).  As  G  can  be  represented  by its defining equations
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  (homogeneous  and  some of them possibly in congruences), we can represent S
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  by  the  defining  equations  of  G; indeed S is just the set of nonnegative
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  solutions  of  this system of equations. We can represent the equations as a
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  list  of lists of integers, all with the same length. Every list is a row of
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  the  matrix of coefficients of the system of equations. For the equations in
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  congruences, if we arrange them so that they are the first ones in the list,
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  we  provide  the  corresponding  moduli  in  a  list.  So  for instance, the
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  equations  x+y≡ 0mod 2, x-2y=0 will be represented as [[1,1],[1,-2]] and the
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  moduli [2].
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  As happens with numerical semigroups, there are different ways to specify an
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  affine  semigroup S, namely, by means of a system of generators, a system of
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  homogeneous  linear  Diophantine equations or a system of homogeneous linear
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  Diophantine  inequalities, just to mention some. In this section we describe
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  functions  that  may  be  used  to  specify, in one of these ways, an affine
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  semigroup in GAP.
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  11.1-1 AffineSemigroupByGenerators
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  AffineSemigroupByGenerators( List )  function
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  AffineSemigroup( String, List )  function
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  List  is  a list of n-tuples of nonnegative integers, if the semigroup to be
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  created  is  n-dimensional.  The  n-tuples  may  be  given as a list or by a
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  sequence  of individual elements. The output is the affine semigroup spanned
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  by List.
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  String  does  not  need  to  be  present.  When  it  is  present, it must be
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  "generators" and List must be a list, not a sequence of individual elements.
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    Example  
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    gap> s1 := AffineSemigroupByGenerators([1,3],[7,2],[1,5]);
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    <Affine semigroup in 2 dimensional space, with 3 generators>
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    gap> s2 := AffineSemigroupByGenerators([[1,3],[7,2],[1,5]]);;
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    gap> s3 := AffineSemigroup("generators",[[1,3],[7,2],[1,5]]);;
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    gap> s4 := AffineSemigroup([1,3],[7,2],[1,5]);;
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    gap> s5 := AffineSemigroup([[1,3],[7,2],[1,5]]);;
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    gap> Length(Set([s1,s2,s3,s4,s5]));
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    1
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  11.1-2 AffineSemigroupByEquations
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  AffineSemigroupByEquations( List )  function
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  AffineSemigroup( String, List )  function
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  List  is  a  list  with  two  components. The first represents a matrix with
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  integer coefficients, say A=(a_ij), and so it is a list of lists of integers
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  all  with  the  same  length.  The  second  component  is a list of positive
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  integers, say d=(d_i), which may be empty. The list d must be of length less
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  than or equal to the length of A (number of rows of A).
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  The  output  is  the  full semigroup of nonnegative integer solutions to the
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  system of homogeneous equations
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  a_11x_1+⋯+a_1nx_n≡ 0mod d_1,
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  ⋯
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  a_k1x_1+⋯+a_knx_n≡ 0mod d_k,
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  a_k+1 1x_1+⋯ +a_k+1 n=0,
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  ⋯
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  a_m1x_1+⋯+a_mnx_n=0.
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  If d is empty, then there will be no equations in congruences.
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  As  pointed  at  the  beginning  of  the section, the equations x+y≡ 0mod 2,
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  x-2y=0  will  be  represented  as A equal to [[1,1],[1,-2]] and the moduli d
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  equal to [2].
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  In the second form, String must be "equations".
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    Example  
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    gap> s1 := AffineSemigroupByEquations([[[-2,1]],[3]]);
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    <Affine semigroup>
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    gap> s2 := AffineSemigroup("equations",[[[1,1]],[3]]);
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    <Affine semigroup>
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    gap> s1=s2;
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    true
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  11.1-3 AffineSemigroupByInequalities
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  AffineSemigroupByInequalities( List )  function
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  AffineSemigroup( String, List )  function
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  List  is  a  list  of  lists (a matrix) of integers that represents a set of
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  inequalities.
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  Returns the (normal) affine semigroup of nonegative integer solutions of the
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  system of inequalities List× Xge 0.
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  In the second form, String must be "inequalities".
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    Example  
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    gap> a1:=AffineSemigroupByInequalities([[2,-1],[-1,3]]);
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    <Affine semigroup>
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    gap> a2:=AffineSemigroup("inequalities",[[2,-1],[-1,3]]);
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    <Affine semigroup>
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    gap> a1=a2;
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    true
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  11.1-4 Generators
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  Generators( S )  function
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  GeneratorsOfAffineSemigroup( S )  function
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  S is an affine semigroup, the output is a system of generators.
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    Example  
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    gap> a:=AffineSemigroup([[1,0],[0,1],[1,1]]);
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    <Affine semigroup in 2 dimensional space, with 3 generators>
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    gap> Generators(a);
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    [ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ] ]
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  11.1-5 MinimalGenerators
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  MinimalGenerators( S )  function
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  MinimalGeneratingSystem( S )  function
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  S is an affine semigroup, the output is its system of minimal generators.
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    Example  
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    gap> a:=AffineSemigroup([[1,0],[0,1],[1,1]]);
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    <Affine semigroup in 2 dimensional space, with 3 generators>
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      gap> MinimalGenerators(a);
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      [ [ 0, 1 ], [ 1, 0 ] ]
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  11.1-6 AsAffineSemigroup
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  AsAffineSemigroup( S )  function
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  S is a numerical semigroup, the output is S regarded as an affine semigroup.
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    Example  
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    gap> s:=NumericalSemigroup(1310,1411,1546,1601);
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    <Numerical semigroup with 4 generators>
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    gap> MinimalPresentationOfNumericalSemigroup(s);;time;
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    2960
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    gap> a:=AsAffineSemigroup(s);
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    <Affine semigroup in 1 dimensional space, with 4 generators>
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    gap> GeneratorsOfAffineSemigroup(a);
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    [ [ 1310 ], [ 1411 ], [ 1546 ], [ 1601 ] ]
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    gap> MinimalPresentationOfAffineSemigroup(a);;time;
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    237972
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  If we use the package SingularInterface, the speed up is considerable.
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    Example  
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    gap> NumSgpsUseSingularInterface();
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    ...
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    gap> MinimalPresentationOfAffineSemigroup(a);;time;
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    32
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  11.1-7 IsAffineSemigroup
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  IsAffineSemigroup( AS )  attribute
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  IsAffineSemigroupByGenerators( AS )  attribute
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  IsAffineSemigroupByEquations( AS )  attribute
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  IsAffineSemigroupByInequalities( AS )  attribute
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  AS  is  an  affine semigroup and these attributes are available (their names
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  should  be self explanatory). They reflect what is currently known about the
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  semigroup.
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    Example  
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    gap> a1:=AffineSemigroup([[3,0],[2,1],[1,2],[0,3]]);
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    <Affine semigroup in 2 dimensional space, with 4 generators>
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    gap> IsAffineSemigroupByEquations(a1);
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    false
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    gap> IsAffineSemigroupByGenerators(a1);
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    true
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  11.1-8 BelongsToAffineSemigroup
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  BelongsToAffineSemigroup( v, a )  function
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  \in( v, a )  operation
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  v  is a list of nonnegative integers and a an affine semigroup. Returns true
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  if the vector is in the semigroup, and false otherwise.
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  If  the  semigroup  is  full and its equations are known (either because the
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  semigroup  was  defined  by  equations,  or  because  the  user  has  called
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  IsFullAffineSemgiroup(a)  and  the  output  was  true),  then  membership is
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  performed  by  evaluating  v  in  the  equations.  The same holds for normal
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  semigroups and its defining inequalities.
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   v in a can be used for short.
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    Example  
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    gap> a:=AffineSemigroup([[2,0],[0,2],[1,1]]);;
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    gap> BelongsToAffineSemigroup([5,5],a);
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    true
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    gap> BelongsToAffineSemigroup([1,2],a);
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    false
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    gap> [5,5] in a;
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    true
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    gap> [1,2] in a;
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    false
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  11.1-9 IsFull
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  IsFull( S )  property
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  IsFullAffineSemigroup( S )  property
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  s is an affine semigroup.
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  Returns  true  if  the  semigroup is full, false otherwise. The semigroup is
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  full  if whenever a,b∈ S and b-a∈ N^k, then a-b∈ S, where k is the dimension
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  of S.
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  If the semigroup is full, then its equations are stored in the semigroup for
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  further use.
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    Example  
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    gap> a:=AffineSemigroup("equations",[[[1,1,1],[0,0,2]],[2,2]]);;
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    gap> IsFullAffineSemigroup(a);
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    true
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  11.1-10 HilbertBasisOfSystemOfHomogeneousEquations
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  HilbertBasisOfSystemOfHomogeneousEquations( ls, m )  operation
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  ls  is a list of lists of integers and m a list of integers. The elements of
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  ls  represent  the  rows  of  a matrix A. The output is a minimal generating
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  system  (Hilbert  basis)  of the set of nonnegative integer solutions of the
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  system  Ax=0 where the k first equations are in the congruences modulo m[i],
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  with k the length of m.
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  If  the  package NormalizInterface has not been loaded, then Contejean-Devie
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  algorithm  is  used  [CD94]  instead  (if  this is the case, congruences are
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  treated as in [RGS98]).
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    Example  
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    gap> HilbertBasisOfSystemOfHomogeneousEquations([[1,0,1],[0,1,-1]],[2]);
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    [ [ 0, 2, 2 ], [ 1, 1, 1 ], [ 2, 0, 0 ] ]
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  If  C  is a pointed cone (a cone in Q^k not containing lines and 0∈ C), then
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  S=C∩  N^k  is  an  affine  semigroup  (known as normal affine semigroup). So
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  another  way  to  give  an  affine  semigroup  is  by  a  set of homogeneous
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  inequalities,  and  we can represent these inequalities by its coefficients.
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  If  we  put  them  in  a  matrix  S can be defined as the set of nonnegative
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  integer solutions to Ax ge 0.
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  11.1-11 HilbertBasisOfSystemOfHomogeneousInequalities
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  HilbertBasisOfSystemOfHomogeneousInequalities( ls )  operation
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  ls  is a list of lists of integers. The elements of ls represent the rows of
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  a matrix A. The output is a minimal generating system (Hilbert basis) of the
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  set of nonnegative integer solutions to Axge 0.
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  If  the  package NormalizInterface has not been loaded, then Contejean-Devie
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  algorithm is used [CD94] instead (the use of slack variables is described in
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  [CAGGB02]).
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    Example  
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    gap> HilbertBasisOfSystemOfHomogeneousInequalities([[2,-3],[0,1]]);
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    [ [ 1, 0 ], [ 2, 1 ], [ 3, 2 ] ]
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  11.1-12 EquationsOfGroupGeneratedBy
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  EquationsOfGroupGeneratedBy( M )  function
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  M  is  a  matrix of integers. The output is a pair [A,m] that represents the
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  set  of  defining  equations  of  the  group spanned by the rows of M: Ax=0∈
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  Z_n_1× ⋯ × Z_n_t× Z^k, with m=[n_1,...,n_t].
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    Example  
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    gap> EquationsOfGroupGeneratedBy([[1,2,0],[2,-2,2]]);
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    [ [ [ 0, 0, -1 ], [ -2, 1, 3 ] ], [ 2 ] ]
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  11.1-13 BasisOfGroupGivenByEquations
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  BasisOfGroupGivenByEquations( A, m )  function
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  A  is  a matrix of integers and m is a list of positive integers. The output
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  is  a  basis  for  the group with defining equations Ax=0∈ Z_n_1× ⋯ × Z_n_t×
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  Z^k, with m=[n_1,...,n_t].
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    Example  
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    gap> BasisOfGroupGivenByEquations([[0,0,1],[2,-1,-3]],[2]);
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    [ [ -1, -2, 0 ], [ -2, 2, -2 ] ]
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  11.2 Gluings of affine semigroups
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  Let  S_1  and S_2 be two affine semigroups with the same dimension generated
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  by  A_1  and A_2, respectively. We say that the affine semigroup S generated
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  by the union of A_1 and A_2 is a gluing of S_1 and S_2 if G(S_1)∩ G(S_2)=d Z
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  (G(⋅) stands for group spanned by) for some d∈ S_1∩ S_2.
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  The algorithm used is explained in [RGS99a].
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  11.2-1 GluingOfAffineSemigroups
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  GluingOfAffineSemigroups( a1, a2 )  function
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  a1,  a2  are  affine semigroups. Determines if they can be glued, and if so,
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  returns the gluing. Otherwise it returns fail.
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    Example  
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    gap> a1:=AffineSemigroup([[2,0],[0,2]]);
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    <Affine semigroup in 2 dimensional space, with 2 generators>
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    gap> a2:=AffineSemigroup([[1,1]]);
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    <Affine semigroup in 2 dimensional space, with 1 generators>
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    gap> GluingOfAffineSemigroups(a1,a2);
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    <Affine semigroup in 2 dimensional space, with 3 generators>
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    gap> Generators(last);
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    [ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ] ]
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  11.3 Presentations of affine semigroups
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  A  minimal presentation of an affine semigroup is defined analogously as for
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  numerical  semigroups.  The user might take into account that generators are
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  stored  in  a  set,  and  thus  might  be arranged in a different way to the
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  initial input.
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  11.3-1 GeneratorsOfKernelCongruence
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  GeneratorsOfKernelCongruence( M )  operation
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  M is matrix with nonnegative integer coefficients. The output is a system of
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  generators of the congruence {(x,y)∣ xM=yM}.
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  The  main  difference  with MinimalPresentationOfAffineSemigroup is that the
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  matrix M can have repeated columns and these are not treated as a set.
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  11.3-2 CanonicalBasisOfKernelCongruence
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  CanonicalBasisOfKernelCongruence( M, Ord )  operation
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  M  is matrix with nonnegative integer coefficients, Ord a term ordering. The
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  output is a canonical bases of the congruence {(x,y)∣ xM=yM} (see [RGS99b]).
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  This corresponds with the exponents of the Gröbner basis of the kernel ideal
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  of the morphism x_i↦ Y^m_i, with m_i the ith row of M.
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  Accepted  term  orderings  are lexicographic (MonomialLexOrdering()), graded
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  lexicographic  (MonomialGrlexOrdering())  and  reversed graded lexicographic
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  (MonomialGrevlexOrdering()).
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    Example  
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    gap> M:=[[3],[5],[7]];;
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    gap> CanonicalBasisOfKernelCongruence(M,MonomialLexOrdering());
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    [ [ [ 0, 7, 0 ], [ 0, 0, 5 ] ], [ [ 1, 0, 1 ], [ 0, 2, 0 ] ],
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      [ [ 1, 5, 0 ], [ 0, 0, 4 ] ], [ [ 2, 3, 0 ], [ 0, 0, 3 ] ],
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      [ [ 3, 1, 0 ], [ 0, 0, 2 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
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    gap> CanonicalBasisOfKernelCongruence(M,MonomialGrlexOrdering());
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    [ [ [ 0, 7, 0 ], [ 0, 0, 5 ] ], [ [ 1, 0, 1 ], [ 0, 2, 0 ] ],
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      [ [ 1, 5, 0 ], [ 0, 0, 4 ] ], [ [ 2, 3, 0 ], [ 0, 0, 3 ] ],
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      [ [ 3, 1, 0 ], [ 0, 0, 2 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
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    gap> CanonicalBasisOfKernelCongruence(M,MonomialGrevlexOrdering());
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    [ [ [ 0, 2, 0 ], [ 1, 0, 1 ] ], [ [ 3, 1, 0 ], [ 0, 0, 2 ] ],
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      [ [ 4, 0, 0 ], [ 0, 1, 1 ] ] ]
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  11.3-3 GraverBasis
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  GraverBasis( M )  operation
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  M is matrix with integer coefficients. The output is a Graver basis for M.
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    Example  
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    gap> gr:=GraverBasis([[3,5,7]]);
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    [ [ -7, 0, 3 ], [ -5, 3, 0 ], [ -4, 1, 1 ], [ -3, -1, 2 ], [ -2, -3, 3 ],
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      [ -1, -5, 4 ], [ -1, 2, -1 ], [ 0, -7, 5 ], [ 0, 7, -5 ], [ 1, -2, 1 ],
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      [ 1, 5, -4 ], [ 2, 3, -3 ], [ 3, 1, -2 ], [ 4, -1, -1 ], [ 5, -3, 0 ],
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      [ 7, 0, -3 ] ]
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  11.3-4 MinimalPresentationOfAffineSemigroup
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403
  MinimalPresentationOfAffineSemigroup( a )  operation
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  MinimalPresentation( a )  operation
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406
  a is an affine semigroup. The output is a minimal presentation for a.
407
  
408
  There  are  four  methods  implemented  for  this function, depending on the
409
  packages  loaded.  All of them use elimination, and Herzog's correspondence,
410
  computing the kernel of a ring homomorphism ([Her70]). The fastest procedure
411
  is  achieved  when  SingularInterface  is  loaded, followed by Singular. The
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  procedure  that  does  not  use  external packages uses internal GAP Gröbner
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  basis  computations  and  thus  it  is  slower.  Also in this case, from the
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  Gröbner  basis,  a  minimal set of generating binomials must be refined, and
415
  for  this  Rclasses  are  used  (if  NormalizInterface  is  loaded, then the
416
  factorizations  are  faster).  The  4ti2  implementation  uses 4ti2 internal
417
  Gröbner bases and factorizations are done via zsolve.
418
  
419
    Example  
420
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
421
    gap> MinimalPresentationOfAffineSemigroup(a);
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    [ [ [ 1, 0, 1 ], [ 0, 2, 0 ] ] ]
423
    gap> GeneratorsOfAffineSemigroup(a);
424
    [ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ] ]
425
  
426
  
427
  11.3-5 BettiElementsOfAffineSemigroup
428
  
429
  BettiElementsOfAffineSemigroup( a )  operation
430
  BettiElements( a )  operation
431
  
432
  a  is  an  affine  semigroup.  The  output is the set of Betti elements of a
433
  (defined as for numerical semigroups).
434
  
435
  This function relies on the computation of a minimal presentation.
436
  
437
    Example  
438
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
439
    gap> BettiElementsOfAffineSemigroup(a);
440
    [ [ 2, 2 ] ]
441
  
442
  
443
  11.3-6 ShadedSetOfElementInAffineSemigroup
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445
  ShadedSetOfElementInAffineSemigroup( v, a )  function
446
  
447
  a  is an affine semigroup and v is an element in a. This is a translation to
448
  affine semigroups of ShadedSetOfElementInNumericalSemigroup (4.1-5).
449
  
450
  11.3-7 IsGeneric
451
  
452
  IsGeneric( a )  property
453
  IsGenericAffineSemigroup( a )  property
454
  
455
  a is an affine semigroup.
456
  
457
  The same as IsGenericNumericalSemigroup (4.2-2) but for affine semigroups.
458
  
459
  11.3-8 IsUniquelyPresentedAffineSemigroup
460
  
461
  IsUniquelyPresentedAffineSemigroup( a )  property
462
  
463
  a is an affine semigroup.
464
  
465
  The  same  as  IsUniquelyPresentedNumericalSemigroup  (4.2-1) but for affine
466
  semigroups.
467
  
468
  11.3-9 PrimitiveElementsOfAffineSemigroup
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470
  PrimitiveElementsOfAffineSemigroup( a )  operation
471
  
472
  a  is  an affine semigroup. The output is the set of primitive elements of a
473
  (defined as for numerical semigroups).
474
  
475
  This  function  has  three implementations (methods), one using Graver basis
476
  via   the   Lawrence  lifting  of  a  and  the  other  (much  faster)  using
477
  NormalizInterface. Also a 4ti2 version using its Graver basis computation is
478
  provided.
479
  
480
    Example  
481
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
482
    gap> PrimitiveElementsOfAffineSemigroup(a);
483
    [ [ 0, 2 ], [ 1, 1 ], [ 2, 0 ], [ 2, 2 ] ]
484
  
485
  
486
  
487
  11.4 Factorizations in affine semigroups
488
  
489
  The invariants presented here are defined as for numerical semigroups.
490
  
491
  As with presentations, the user should take into account that generators are
492
  stored  in  a  set,  and  thus  might  be arranged in a different way to the
493
  initial input.
494
  
495
  11.4-1 FactorizationsVectorWRTList
496
  
497
  FactorizationsVectorWRTList( v, ls )  operation
498
  
499
  v is a list of nonnegative integers and ls is a list of lists of nonnegative
500
  integers.  The output is set of factorizations of v in terms of the elements
501
  of ls.
502
  
503
  If no extra package is loaded, then factorizations are computed recursively;
504
  and  thus slowly. If NormalizInterface is loaded, then a system of equations
505
  is   solved   with   Normaliz,  and  the  performance  is  much  better.  If
506
  4ti2Interface  is  loaded  instead, then factorizations are calculated using
507
  zsolve command of 4ti2.
508
  
509
    Example  
510
    gap> FactorizationsVectorWRTList([5,5],[[2,0],[0,2],[1,1]]);
511
    [ [ 2, 2, 1 ], [ 1, 1, 3 ], [ 0, 0, 5 ] ]
512
  
513
  
514
  11.4-2 ElasticityOfAffineSemigroup
515
  
516
  ElasticityOfAffineSemigroup( a )  operation
517
  
518
  a  is an affine semigroup. The output is the elasticity of a (defined as for
519
  numerical semigroups).
520
  
521
  The  procedure  used  is  based  on  [Phi10],  where  it  is  shown that the
522
  elasticity  can  be  computed  by  using  circuits.  The  set of circuits is
523
  calculated using [ES96].
524
  
525
    Example  
526
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
527
    gap> ElasticityOfAffineSemigroup(a);
528
    1
529
  
530
  
531
  11.4-3 DeltaSetOfAffineSemigroup
532
  
533
  DeltaSetOfAffineSemigroup( a )  function
534
  
535
  a  is  an affine semigroup. The output is the Delta set of a (defined as for
536
  numerical semigroups). The the procedure used is explained in [GSOW17].
537
  
538
    Example  
539
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
540
    gap> DeltaSetOfAffineSemigroup(a);
541
    [  ]
542
    gap> s:=NumericalSemigroup(10,13,15,47);;
543
    gap> a:=AsAffineSemigroup(s);;
544
    gap> DeltaSetOfAffineSemigroup(a);
545
    [ 1, 2, 3, 5 ]
546
  
547
  
548
  11.4-4 CatenaryDegreeOfAffineSemigroup
549
  
550
  CatenaryDegreeOfAffineSemigroup( a )  function
551
  
552
  a is an affine semigroup. The output is the catenary degree of a (defined as
553
  for numerical semigroups).
554
  
555
    Example  
556
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
557
    gap> CatenaryDegreeOfAffineSemigroup(a);
558
    2
559
  
560
  
561
  11.4-5 EqualCatenaryDegreeOfAffineSemigroup
562
  
563
  EqualCatenaryDegreeOfAffineSemigroup( a )  function
564
  
565
  a  is  an  affine  semigroup.  The  output is the equal catenary degree of a
566
  (defined as for numerical semigroups).
567
  
568
  This function relies on the results presented in [GSOSRN13].
569
  
570
  11.4-6 HomogeneousCatenaryDegreeOfAffineSemigroup
571
  
572
  HomogeneousCatenaryDegreeOfAffineSemigroup( a )  function
573
  
574
  a is an affine semigroup. The output is the homogeneous catenary degree of a
575
  (defined as for numerical semigroups).
576
  
577
  This function is based on [GSOSRN13].
578
  
579
  11.4-7 MonotoneCatenaryDegreeOfAffineSemigroup
580
  
581
  MonotoneCatenaryDegreeOfAffineSemigroup( a )  function
582
  
583
  a  is  an  affine semigroup. The output is the monotone catenary degree of a
584
  (defined as for numerical semigroups), computed as explained in [Phi10].
585
  
586
    Example  
587
    gap> a:=AffineSemigroup("inequalities",[[2,-1],[-1,3]]);
588
    <Affine semigroup>
589
    gap> GeneratorsOfAffineSemigroup(a);
590
    [ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ], [ 3, 1 ] ]
591
    gap> CatenaryDegreeOfAffineSemigroup(a);
592
    3
593
    gap> EqualCatenaryDegreeOfAffineSemigroup(a);
594
    2
595
    gap> HomogeneousCatenaryDegreeOfAffineSemigroup(a);
596
    3
597
    gap> MonotoneCatenaryDegreeOfAffineSemigroup(a);
598
    3
599
  
600
  
601
  11.4-8 TameDegreeOfAffineSemigroup
602
  
603
  TameDegreeOfAffineSemigroup( a )  operation
604
  
605
  a is an affine semigroup. The output is the tame degree of a (defined as for
606
  numerical  semigroups).  If  a  is  given by equations (or its equations are
607
  known), then the procedure explained in [GSOW17] is used.
608
  
609
    Example  
610
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
611
    gap> TameDegreeOfAffineSemigroup(a);
612
    2
613
  
614
  
615
  11.4-9 OmegaPrimalityOfElementInAffineSemigroup
616
  
617
  OmegaPrimalityOfElementInAffineSemigroup( v, a )  operation
618
  
619
  v is a list of nonnegative integers and a is an affine semigroup. The output
620
  is the omega primality of a (defined as for numerical semigroups). Returns 0
621
  if the element is not in the semigroup.
622
  
623
  The  implementation  of this procedure is performed as explained in [BGSG11]
624
  (also,  if  the semigroup has defining equations, then it takes advantage of
625
  this fact as explained in this reference).
626
  
627
    Example  
628
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
629
    gap> OmegaPrimalityOfElementInAffineSemigroup([5,5],a);
630
    6
631
  
632
  
633
  11.4-10 OmegaPrimalityOfAffineSemigroup
634
  
635
  OmegaPrimalityOfAffineSemigroup( a )  function
636
  
637
  a is an affine semigroup. The output is the omega primality of a (defined as
638
  for numerical semigroups).
639
  
640
    Example  
641
    gap> a:=AffineSemigroup([2,0],[0,2],[1,1]);;
642
    gap> OmegaPrimalityOfAffineSemigroup(a);
643
    2
644
  
645
  
646

647