chap12.txt

  
  12 Good semigroups
  
  We will only cover here good semigroups of N^2.
  
  A good semigroup S is a submonoid of N^2, with the following properties.
  
  (G1) It is closed under infimums (minimum componentwise).
  
  (G2)  if a, b ∈ M and a_i = b_i for some i ∈ {1, 2}, then there exists c ∈ M
  such that c_i > a_i = b_i and c_j = min{a_j,b_j}, with j∈{1,2}∖ {i}.
  
  (G3) there exists C∈N^n such that C+N^n⊆ S.
  
  Value  semigroups  of  algebroid branches are good semigroups, but there are
  good  semigroups that are not of this form. Since good semigroups are closed
  under  infimums, if C_1 and C_2 fulfill C_i+N^n⊆ S, then C_1∧ C_2+N^n⊆ S. So
  there  is  a minimum C fulfilling C+N^n⊆ S, which is called the conductor of
  S.
  
  The contents of this chapter are described in [DGSM16].
  
  
  12.1 Defining good semigroups
  
  12.1-1 IsGoodSemigroup
  
  IsGoodSemigroup( S )  function
  
  Detects if S is an object of type good semigroup.
  
  12.1-2 NumericalSemigroupDuplication
  
  NumericalSemigroupDuplication( S, E )  function
  
  S  is  a numerical semigroup and E is an ideal of S with E⊆ S. The output is
  S⋈ E= D∪ (E× E)∪{ a∧ b∣ a∈ D, b∈ E× E}, where D={(s,s)∣ s∈ S}.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> l:=Cartesian([1..11],[1..11]);;
    gap> Intersection(dup,l);
    [ [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
      [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ],
      [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ],
      [ 11, 9 ], [ 11, 11 ] ]
    gap> [384938749837,349823749827] in dup;
    true
  
  
  12.1-3 AmalgamationOfNumericalSemigroups
  
  AmalgamationOfNumericalSemigroups( S, E, b )  function
  
  S is a numerical semigroup, E is an ideal of a numerical semigroup T with E⊆
  T, and b is an integer such that multiplication by b is a morphism from S to
  T, say g. The output is S⋈^g E= D∪(g^-1(E)× E)∪ {a∧ b∣ a∈ D, b∈ g^-1(E)× E},
  where D={(s,b s)∣ s∈ S}.
  
    Example  
    gap> s:=NumericalSemigroup(2,3);;
    gap> t:=NumericalSemigroup(3,4);;
    gap> e:=3+t;;
    gap> dup:=AmalgamationOfNumericalSemigroups(s,e,2);;
    gap> [2,3] in dup;
    true
  
  
  12.1-4 CartesianProductOfNumericalSemigroups
  
  CartesianProductOfNumericalSemigroups( S, T )  function
  
  S  and  T  are  numerical  semigroups.  The  output is S× T, which is a good
  semigroup.
  
    Example  
    gap> s:=NumericalSemigroup(2,3);;
    gap> t:=NumericalSemigroup(3,4);;
    gap> IsGoodSemigroup(CartesianProductOfNumericalSemigroups(s,t));
    true
  
  
  12.1-5 GoodSemigroup
  
  GoodSemigroup( X, C )  function
  
  X  is  a list of points with nonnegative integer coordinates and C is a pair
  of  nonnegative  integer  (a list with two elements). If M is the affine and
  infimum  closure of X, decides if it is a good semigroup, and if so, outputs
  it.
  
    Example  
    gap> G:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];
    [ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],
      [ 25, 12 ], [ 25, 16 ] ]
    gap> C:=[25,27];
    [ 25, 27 ]
    gap> GoodSemigroup(G,C);
    <Good semigroup>
  
  
  
  12.2 Notable elements
  
  12.2-1 BelongsToGoodSemigroup
  
  BelongsToGoodSemigroup( v, S )  operation
  \in( v, S )  operation
  
  S  is  a good semigroup and v is a pair of integers. The output is true if v
  is  in S, and false otherwise. Other ways to use this operation are \in(v,S)
  and v in S.
  
    Example  
    gap> s:=NumericalSemigroup(2,3);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);;
    gap> BelongsToGoodSemigroup([2,2],dup);
    true
    gap> [2,2] in dup;
    true
    gap> [3,2] in dup;
    false
  
  
  12.2-2 Conductor
  
  Conductor( S )  function
  ConductorOfGoodSemigroup( S )  function
  
  S is a good semigroup. The output is its conductor.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> Conductor(dup);
    [ 11, 11 ]
    gap> ConductorOfGoodSemigroup(dup);
    [ 11, 11 ]
  
  
  12.2-3 SmallElements
  
  SmallElements( S )  function
  SmallElementsOfGoodSemigroup( S )  function
  
  S is a good semigroup. The output is its set of small elements, that is, the
  elements  smaller  than  its  conductor  with  respect  to the usual partial
  ordering.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> SmallElementsOfGoodSemigroup(dup);
    [ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ],
      [ 6, 10 ], [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ],
      [ 9, 9 ], [ 9, 10 ], [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ],
      [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
  
  
  12.2-4 RepresentsSmallElementsOfGoodSemigroup
  
  RepresentsSmallElementsOfGoodSemigroup( X )  function
  
  X  is  a list of points in the nonnegative orthant of the plane with integer
  coordinates. Determines if it represents the set of small elements of a good
  semigroup.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> SmallElementsOfGoodSemigroup(dup);
    [ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
      [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],
      [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
    gap> RepresentsSmallElementsOfGoodSemigroup(last);
    true
  
  
  12.2-5 GoodSemigroupBySmallElements
  
  GoodSemigroupBySmallElements( X )  function
  
  X  is  a list of points in the nonnegative orthant of the plane with integer
  coordinates. Determines if it represents the set of small elements of a good
  semigroup,  and  then  outputs  the  good semigroup having X as set of small
  elements.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> SmallElementsOfGoodSemigroup(dup);
    [ [ 0, 0 ], [ 3, 3 ], [ 5, 5 ], [ 6, 6 ], [ 6, 7 ], [ 6, 8 ], [ 6, 9 ], [ 6, 10 ],
      [ 6, 11 ], [ 7, 6 ], [ 7, 7 ], [ 8, 6 ], [ 8, 8 ], [ 9, 6 ], [ 9, 9 ], [ 9, 10 ],
      [ 9, 11 ], [ 10, 6 ], [ 10, 9 ], [ 10, 10 ], [ 11, 6 ], [ 11, 9 ], [ 11, 11 ] ]
    gap> G:=GoodSemigroupBySmallElements(last);
    <Good semigroup>
    gap> dup=G;
    true
  
  
  12.2-6 MaximalElementsOfGoodSemigroup
  
  MaximalElementsOfGoodSemigroup( S )  attribute
  
  S is a good semigroup. The output is the set of elements (x,y) of S with the
  following  property:  there  is  no  other element (x',y') in S with (x,y)le
  (x',y') sharing a coordinate with (x,y).
  
    Example  
    gap> G:=[[4,3],[7,13],[11,17]];;
    gap> g:=GoodSemigroup(G,[11,17]);;
    gap> mx:=MaximalElementsOfGoodSemigroup(g);
    [ [ 0, 0 ], [ 4, 3 ], [ 7, 13 ], [ 8, 6 ] ]
  
  
  12.2-7 IrreducibleMaximalElementsOfGoodSemigroup
  
  IrreducibleMaximalElementsOfGoodSemigroup( S )  attribute
  
  S  is  a  good  semigroup. The output is the set of elements nonzero maximal
  elements  that  cannot be expressed as a sum of two nonzero maximal elements
  of the good semigroup.
  
    Example  
    gap> G:=[[4,3],[7,13],[11,17]];;
    gap> g:=GoodSemigroup(G,[11,17]);;
    gap> IrreducibleMaximalElementsOfGoodSemigroup(g);
    [ [ 4, 3 ], [ 7, 13 ] ]
  
  
  12.2-8 GoodSemigroupByMaximalElements
  
  GoodSemigroupByMaximalElements( S, T, M, C )  function
  
  S  and  T  are  numerical semigroups, M is a list of pairs in S× T. C is the
  conductor, and thus a pair of nonnegative integers. The output is the set of
  elements  of S× T that are not above an element in M, that is, if they share
  a  coordinate  with  an  element in M, then they must be smaller or equal to
  that  element  with  respect  to the usual partial ordering. The output is a
  good semigroup, if M is an correct set of maximal elements.
  
    Example  
    gap> G:=[[4,3],[7,13],[11,17]];;
    gap> g:=GoodSemigroup(G,[11,17]);;
    gap> sm:=SmallElements(g);;
    gap> mx:=MaximalElementsOfGoodSemigroup(g);;
    gap> s:=NumericalSemigroupBySmallElements(Set(sm,x->x[1]));;
    gap> t:=NumericalSemigroupBySmallElements(Set(sm,x->x[2]));;
    gap> Conductor(g);
    [ 11, 15 ]
    gap> gg:=GoodSemigroupByMaximalElements(s,t,mx,[11,15]);
    <Good semigroup>
    gap> gg=g;
    true
  
  
  12.2-9 MinimalGoodGeneratingSystemOfGoodSemigroup
  
  MinimalGoodGeneratingSystemOfGoodSemigroup( S )  function
  
  S  is  a  good  semigroup.  The output is its minimal good generating system
  (which is unique in the local case).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> MinimalGoodGeneratingSystemOfGoodSemigroup(dup);
    [ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ]
  
  
  12.2-10 MinimalGenerators
  
  MinimalGenerators( S )  attribute
  
  This is just a synonym of  MinimalGoodGeneratingSystemOfGoodSemigroup (S).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=6+s;;
    gap> dup:=NumericalSemigroupDuplication(s,e);
    <Good semigroup>
    gap> MinimalGenerators(dup);
    [ [ 3, 3 ], [ 5, 5 ], [ 6, 11 ], [ 7, 7 ], [ 11, 6 ] ]
  
  
  
  12.3 Symmetric semigroups
  
  12.3-1 IsSymmetricGoodSemigroup
  
  IsSymmetricGoodSemigroup( S )  attribute
  IsSymmetric( S )  attribute
  
  S is a good semigroup. Determines if S is a symmetric good semigroup.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=CanonicalIdealOfNumericalSemigroup(s);;
    gap> e:=15+e;;
    gap> dup:=NumericalSemigroupDuplication(s,e);;
    gap> IsSymmetricGoodSemigroup(dup);
    true
  
  
  12.3-2 ArfGoodSemigroupClosure
  
  ArfGoodSemigroupClosure( S )  function
  ArfClosure( S )  operation
  
  S is a good semigroup. Determines the Arf good semigroup closure of S.
  
    Example  
    gap> G:=[[3,3],[4,4],[5,4],[4,6]];
    [ [ 3, 3 ], [ 4, 4 ], [ 5, 4 ], [ 4, 6 ] ]
    gap> C:=[6,6];
    [ 6, 6 ]
    gap> S:=GoodSemigroup(G,C);
    <Good semigroup>
    gap> SmallElements(S);
    [ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ], [ 4, 6 ], [ 5, 4 ], [ 6, 6 ] ]
    gap> A:=ArfGoodSemigroupClosure(S);
    <Good semigroup>
    gap> SmallElements(A);
    [ [ 0, 0 ], [ 3, 3 ], [ 4, 4 ] ]
  
  
  
  12.4 Good ideals
  
  A  relative  ideal I of a relative good semigroup M is a relative good ideal
  if I fulfills conditions (G1) and (G2) of the definition of good semigroup.
  
  12.4-1 GoodIdeal
  
  GoodIdeal( X, S )  function
  
  X  is  a  list  of points with nonnegative integer coordinates and S is good
  semigroup.  Decides  if the closure of X+S under infimums is a relative good
  ideal of S, and if so, outputs it.
  
    Example  
    gap> G:=[[4,3],[7,13],[11,17],[14,27],[15,27],[16,20],[25,12],[25,16]];
    [ [ 4, 3 ], [ 7, 13 ], [ 11, 17 ], [ 14, 27 ], [ 15, 27 ], [ 16, 20 ],
    [ 25, 12 ], [ 25, 16 ] ]
    gap> C:=[25,27];
    [ 25, 27 ]
    gap> g := GoodSemigroup(G,C);
    <Good semigroup>
    gap> i:=GoodIdeal([[2,3]],g);
    <Good ideal of good semigroup>
  
  
  12.4-2 GoodGeneratingSystemOfGoodIdeal
  
  GoodGeneratingSystemOfGoodIdeal( I )  function
  
  I  is  a  good  ideal  of  a good semigroup. The output is a good generating
  system of I.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> d:=NumericalSemigroupDuplication(s,e);;
    gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],d);;
    gap> GoodGeneratingSystemOfGoodIdeal(e);
    [ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ] ]
  
  
  12.4-3 AmbientGoodSemigroupOfGoodIdeal
  
  AmbientGoodSemigroupOfGoodIdeal( I )  function
  
  If I is a good ideal of a good semigroup M, then the output is M. The output
  is a good generating system of I.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> a:=AmalgamationOfNumericalSemigroups(s,e,5);;
    gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],a);;
    gap> a=AmbientGoodSemigroupOfGoodIdeal(e);
    true
  
  
  12.4-4 MinimalGoodGeneratingSystemOfGoodIdeal
  
  MinimalGoodGeneratingSystemOfGoodIdeal( I )  function
  
  I  is  a  good  ideal  of  a  good semigroup. The output is the minimal good
  generating system of I.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> d:=NumericalSemigroupDuplication(s,e);;
    gap> e:=GoodIdeal([[2,3],[3,2],[2,2]],d);;
    gap> MinimalGoodGeneratingSystemOfGoodIdeal(e);
    [ [ 2, 3 ], [ 3, 2 ] ]
  
  
  12.4-5 BelongsToGoodIdeal
  
  BelongsToGoodIdeal( v, I )  operation
  \in( v, I )  operation
  
  I  is  a  good  ideal  of  a good semigroup and v is a pair of integers. The
  output  is  true  if  v is in I, and false otherwise. Other ways to use this
  operation are \in(v,I) and v in I.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> d:=NumericalSemigroupDuplication(s,e);;
    gap> e:=GoodIdeal([[2,3],[3,2]],d);;
    gap> [1,1] in e;
    false
    gap> [2,2] in e;
    true
  
  
  12.4-6 SmallElementsOfGoodIdeal
  
  SmallElementsOfGoodIdeal( I )  function
  SmallElements( I )  function
  
  I  is  a  good  ideal. The output is its set of small elements, that is, the
  elements  smaller  than  its  conductor  and larger than its minimum element
  (with respect to the usual partial ordering).
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> d:=NumericalSemigroupDuplication(s,e);;
    gap> e:=GoodIdeal([[2,3],[3,2]],d);;
    gap> SmallElements(e);
    [ [ 2, 2 ], [ 2, 3 ], [ 3, 2 ], [ 5, 5 ], [ 5, 6 ], [ 6, 5 ], [ 7, 7 ] ]
  
  
  12.4-7 CanonicalIdealOfGoodSemigroup
  
  CanonicalIdealOfGoodSemigroup( S )  function
  
  S is a good semigroup. The output is the canonical ideal of S.
  
    Example  
    gap> s:=NumericalSemigroup(3,5,7);;
    gap> e:=10+s;;
    gap> d:=NumericalSemigroupDuplication(s,e);;
    gap> c:=CanonicalIdealOfGoodSemigroup(d);;
    gap> MinimalGoodGeneratingSystemOfGoodIdeal(c);
    [ [ 0, 0 ], [ 2, 2 ] ]