chap6.txt

  
  6 Usage of IntPic in connection with the numericalsgps package
  
  This  chapter  describes  functions  to  be  used  in  connection  with  the
  numericalsgps  package.  We  found it particularly usefull to gain intuition
  that led to the obtention of the results stated in [DGSRP16].
  
  
  6.1 Tikz code for drawing numerical semigroups
  
  
  6.1-1 Tikz code for drawing numerical semigroups
  
  TikzCodeForNumericalSemigroup( arg )  function
  
  The arguments (at most 4) are:
  
  1   a numerical semigroup
  
  2   (optional) a list whose elements are either
  
            lists of integers or
  
            one   of   the   strings  "pseudo_frobenius",  "small_elements",
              "min_generators",        "frobenius_number",        "conductor",
              "special_gaps",  "fundamental_gaps"  (the  default: used when no
              list is present) or
  
  3   a record whose fields are
  
            func -- a function name
  
            (optional)  argument -- an argument (that may be a function name
              also)
  
  4   (optional) a positive integer -- if it is bigger than the conductor or
        biggest  minimal generator, it indicates the number of cells - 1 to be
        drawn  and  these  are drawn in a single line; otherwise, it indicates
        the maximum number of cells per line.
  
    Example  
    gap> ns1 := NumericalSemigroup(3,5);;
    gap> TikzCodeForNumericalSemigroup(ns1,[[3,4],"pseudo_frobenius"],20);
    "%tikz\n\\begin{tikzpicture}[every node/.style={draw,scale=1pt,\nminimum width\
    =20pt,inner sep=3pt,\nline width=0pt,draw=black}]\n\\matrix[row sep=2pt,column\
     sep=2pt]\n{\\node[]{0};&\n\\node[]{1};&\n\\node[]{2};&\n\\node[fill=red]{3};&\
    \n\\node[fill=red]{4};&\n\\node[]{5};&\n\\node[]{6};&\n\\node[fill=green]{7};&\
    \n\\node[]{8};&\n\\node[]{9};&\n\\node[]{10};&\n\\node[]{11};&\n\\node[]{12};&\
    \n\\node[]{13};&\n\\node[]{14};&\n\\node[]{15};&\n\\node[]{16};&\n\\node[]{17}\
    ;&\n\\node[]{18};&\n\\node[]{19};&\n\\node[]{20};\\\\\n};\n\\end{tikzpicture}\
    \n"
  
  
    Example  
    gap> IP_Splash(TikzCodeForNumericalSemigroup(NumericalSemigroup(7,13,19,23),
    > [[3,4],"small_elements","fundamental_gaps"],20),rec(viewer := "evince"));    
  
  
  
  6.2 Drawing sets of numerical semigroups
  
  
  6.2-1 Set of numerical semigroups
  
  SetOfNumericalSemigroups( arg )  function
  
  This  function is used to produce lists of numerical semigroups with a fixed
  genus  or  Frobenius number. They are filtered and ordered according to some
  criteria.
  
  The argument is a record of options:
  
  1   set -- a record whose possible fields are genus or frobenius
  
  2   (optional)  filter  --  a record whose possible fields are genus, type
        and/or multiplicity and/or frobenius and/or embedding_dimension
  
  3   (optional)  order  --  ("genus",  "type", "multiplicity", "frobenius",
        "embedding_dimension")
  
    Example  
    gap> SetOfNumericalSemigroups(rec(set:=rec(genus:=6),filter:=rec(type:= 2),
    > order:="multiplicity"));
    [ <Numerical semigroup with 3 generators>, 
      <Numerical semigroup with 3 generators>, 
      <Numerical semigroup with 4 generators>, 
      <Numerical semigroup with 5 generators>, 
      <Numerical semigroup with 5 generators>, 
      <Numerical semigroup with 6 generators> ]
    gap> SetOfNumericalSemigroups(rec(set:=rec(genus:=6),filter:=rec(type:= 2),
    > order:="frobenius"));                                                 
    [ <Numerical semigroup with 3 generators>, 
      <Numerical semigroup with 5 generators>, 
      <Numerical semigroup with 5 generators>, 
      <Numerical semigroup with 3 generators>, 
      <Numerical semigroup with 4 generators>, 
      <Numerical semigroup with 6 generators> ]
    gap> List(last,MinimalGeneratingSystem);
    [ [ 3, 10, 11 ], [ 5, 6, 7 ], [ 5, 6, 8 ], [ 3, 8, 13 ], [ 4, 7, 9 ], 
      [ 6, 7, 8, 9, 11 ] ]
  
  
  
  6.2-2 Draw a set of numerical semigroups
  
  DrawSetOfNumericalSemigroups( arg )  function
  
  Produces a single image from the images of a set of numerical semigroups.
  
  The arguments (at most 3) are:
  
  1   a  list  of  numerical  semigroups  (given  as a list or each given as
        argument)
  
  2   (optional)  an  integer  that  (when present) determines the length of
        each line
  
  3   (optional) a record whose fields are
  
            (optional)  splash  -- which (when present) consists of a record
              of options for the Viz Splash function
  
            (optional)  highlights: a list to be passed to the function that
              produces  the tikz code for each individual semigroup (whose aim
              is to say which elements are to be highlighted)
  
    Example  
    gap> ns1 := NumericalSemigroup(3,5);;
    gap> ns2 := NumericalSemigroup(5,7,11);;
    gap> DrawSetOfNumericalSemigroups(ns1,rec(splash:=                          
    > rec(viewer := "evince"),highlights := 
    > ["pseudo_frobenius","small_elements","min_generators"]));
    gap> DrawSetOfNumericalSemigroups(ns1,ns2,rec(splash:= 
    > rec(viewer := "evince"),highlights := 
    > ["pseudo_frobenius","small_elements","min_generators"]));
    gap> DrawSetOfNumericalSemigroups([ns1,ns2],rec(splash:=      
    > rec(viewer := "evince"),highlights :=                    
    > ["small_elements","min_generators"]));                   
  
  
    Example  
    gap> frob := 15;;
    gap> tipo := 2;;
    gap> set := SetOfNumericalSemigroups(rec(set := rec(frobenius := frob),
    > filter := rec(type:= tipo),order := "embedding_dimension"));;
    gap> DrawSetOfNumericalSemigroups(set,rec(splash:= rec(viewer := "evince"),
    > highlights := ["small_elements","min_generators",
    > rec(func:= "ForcedIntegersForPseudoFrobenius", 
    > argument := "PseudoFrobeniusOfNumericalSemigroup")]));
  
  
  The  following  example  helps in the understanding of the colors present in
  the first line of previous figure.
  
    Example  
    gap> ns := NumericalSemigroup(4,9,19);;
    gap> SmallElements(ns);
    [ 0, 4, 8, 9, 12, 13, 16 ]
    gap> MinimalGeneratingSystem(ns);
    [ 4, 9, 19 ]
    gap> ForcedIntegersForPseudoFrobenius(PseudoFrobeniusOfNumericalSemigroup(ns));
    [ [ 1, 2, 3, 5, 6, 7, 14, 15 ], [ 0, 8, 9, 12, 13, 16 ] ]