chap3.txt

  
  3 Drawings of semigroups
  
  There  are  some  pictures  that  may  give  a  lot  of  information about a
  semigroup.  This  is  the  case of the egg-box picture of the D-classes, the
  right  Cayley  graph  of  a finite monoid and the Schützenberger graphs of a
  finite inverse monoid.
  
  
  3.1 Drawing the D-class of an element of a semigroup
  
  3.1-1 DrawDClassOfElement
  
  DrawDClassOfElement( arg )  function
  
  This  function  uses DotForDrawingDClassOfElement (3.1-2) to compute the dot
  code  to  produce  the image that is then displayed. It takes as arguments a
  semigroup  followed  by  a transformation which is the element whose D-class
  will  be  drawn.  Optionally we can then specify n lists of elements and the
  elements  of  each  list will be drawn in different colours. Finally, if the
  last   argument   is  the  integer  1  then  the  elements  will  appear  as
  transformations,  otherwise  they will appear as words. The idempotents will
  be marked with a * before them.
  
  This  last  optional argument may also be the integer 2 and in this case the
  elements   will   appear   as  integers,  where  i  represents  the  element
  Elements(S)[i].
  
    Example  
    gap> DrawDClassOfElement(poi3, Transformation([1,4,3,4]));
    gap> DrawDClassOfElement(poi3, Transformation([1,4,3,4]),1);
    gap> DrawDClassOfElement(poi3, Transformation([1,4,3,4]),
     [Transformation( [ 2, 3, 4, 4 ] )],1);
    gap> DrawDClassOfElement(poi3, Transformation([1,4,3,4]),
     [Transformation( [ 2, 3, 4, 4 ] ), Transformation( [ 2, 4, 3, 4 ] )],
     [Transformation( [ 2, 4, 3, 4 ] )],1);
    gap> DrawDClassOfElement(poi3, Transformation([1,4,3,4]), 
     [Transformation( [ 2, 4, 3, 4 ] )],"Dclass",1);
          
  
  
  This  is  the  image  produced by the first command in the previous example:
  This  is  the  image produced by the fourth command in the previous example:
  This is the image produced by the last command in the previous example:
  
  3.1-2 DotForDrawingDClassOfElement
  
  DotForDrawingDClassOfElement( arg )  function
  
  This  function  computes  the dot code that can be used to produce a drawing
  for  the D-class of an element of a semigroup. This can be done by using the
  function  DrawDClassOfElement (3.1-1) (if the system is properly configured)
  or  by  the  user in some independent way. The arguments and options are the
  same than those of DrawDClassOfElement (3.1-1).
  
    Example  
          gap> DotForDrawingDClassOfElement(poi3,Transformation([1,4,3,4]));
    "digraph  DClassOfElement {\ngraph [center=yes,ordering=out];\nnode [shape=pla\
    intext];\nedge [color=cornflowerblue,arrowhead=none];\n1 [label=<\n<TABLE BORD\
    ER=\"0\" CELLBORDER=\"0\" CELLPADDING=\"0\" CELLSPACING=\"0\" PORT=\"1\">\n<TR\
    ><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR><TD BGCOLOR=\"white\" BORDER=\"\
    0\">*abc</TD></TR>\n</TABLE></TD><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR\
    ><TD BGCOLOR=\"white\" BORDER=\"0\">a</TD></TR>\n</TABLE></TD><TD BORDER=\"0\"\
    ><TABLE CELLSPACING=\"0\"><TR><TD BGCOLOR=\"white\" BORDER=\"0\">ab</TD></TR>\
    \n</TABLE></TD></TR>\n<TR><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR><TD BG\
    COLOR=\"white\" BORDER=\"0\">bc</TD></TR>\n</TABLE></TD><TD BORDER=\"0\"><TABL\
    E CELLSPACING=\"0\"><TR><TD BGCOLOR=\"white\" BORDER=\"0\">*bca</TD></TR>\n</T\
    ABLE></TD><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR><TD BGCOLOR=\"white\" \
    BORDER=\"0\">b</TD></TR>\n</TABLE></TD></TR>\n<TR><TD BORDER=\"0\"><TABLE CELL\
    SPACING=\"0\"><TR><TD BGCOLOR=\"white\" BORDER=\"0\">c</TD></TR>\n</TABLE></TD\
    ><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR><TD BGCOLOR=\"white\" BORDER=\"\
    0\">ca</TD></TR>\n</TABLE></TD><TD BORDER=\"0\"><TABLE CELLSPACING=\"0\"><TR><\
    TD BGCOLOR=\"white\" BORDER=\"0\">*cab</TD></TR>\n</TABLE></TD></TR>\n</TABLE>\
    >];\n}\n"
          
  
  
  By  using  Print  (or PrinTo, if one wants to print to a file) the string is
  displayed as follows:
  
    Example  
    gap> Print(last);
    digraph  DClassOfElement {
    graph [center=yes,ordering=out];
    node [shape=plaintext];
    edge [color=cornflowerblue,arrowhead=none];
    1 [label=<
    <TABLE BORDER="0" CELLBORDER="0" CELLPADDING="0" CELLSPACING="0" PORT="1">
    <TR><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BORDER="0">*\
    abc</TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">a</TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">ab</TD></TR>
    </TABLE></TD></TR>
    <TR><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BORDER="0">b\
    c</TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">*bca</TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">b</TD></TR>
    </TABLE></TD></TR>
    <TR><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BORDER="0">c\
    </TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">ca</TD></TR>
    </TABLE></TD><TD BORDER="0"><TABLE CELLSPACING="0"><TR><TD BGCOLOR="white" BOR\
    DER="0">*cab</TD></TR>
    </TABLE></TD></TR>
    </TABLE>>];
    }
  
  
  
  3.2 Drawing the D-classes of a semigroup
  
  3.2-1 DrawDClasses
  
  DrawDClasses( arg )  function
  
  This function is similar to the previous one, except that this one draws all
  the  D-classes  of  the semigroup given as the first argument. It then takes
  optionally  n  lists of elements and the elements of each list will be drawn
  in  different colours. It also accepts, as an optional argument, the integer
  1,   to   specify   whether   the  elements  will  appear  as  words  or  as
  transformations  as in the previous function. The idempotents will be marked
  with a * before them.
  
  The dot code is computed by DotForDrawingDClasses (3.2-2).
  
  This  last  optional argument may also be the integer 2 and in this case the
  elements   will   appear   as  integers,  where  i  represents  the  element
  Elements(S)[i].
  
    Example  
    gap> DrawDClasses(poi3);
    gap> DrawDClasses(poi3, [Transformation( [ 2, 3, 4, 4 ] ),
      Transformation( [ 2, 4, 3, 4 ] )],
      [Transformation( [ 2, 4, 3, 4 ] )],1);
          
  
  
  This  is  the  image  produced by the first command in the previous example:
  This is the image produced by the second command in the previous example:
  
  3.2-2 DotForDrawingDClasses
  
  DotForDrawingDClasses( arg )  function
  
  This  function  computes  the dot code that can be used to produce a drawing
  for  the D-class of an element of a semigroup. This can be done by using the
  function  DrawDClasses  (3.2-1) (if the system is properly configured) or by
  the  user  in  some  independent way. The arguments and options are the same
  than those of DrawDClasses (3.2-1).
  
    Example  
    gap> Print(DotForDrawingDClasses(poi3));
    digraph  DClasses {
    graph [center=yes,ordering=out];
    node [shape=plaintext];
    edge [color=cornflowerblue,arrowhead=none];
    ## ... many more lines ...
    </TABLE></TD></TR>
    </TABLE>>];
    4:4 -> 3:3;
    3:3 -> 2:2;
    2:2 -> 1:1;
    }
          
  
  
  
  3.3 Cayley graphs
  
  3.3-1 DrawRightCayleyGraph
  
  DrawRightCayleyGraph( S )  function
  
  Draws the right Cayley graph of a finite monoid or semigroup S.
  
  3.3-2 DrawCayleyGraph
  
  DrawCayleyGraph( S )  function
  
  This function is a synonym of DrawRightCayleyGraph (3.3-1).
  
  For  example,  the command DrawCayleyGraph(b21); would produce the following
  image  (where  state  i  represents  the element Elements(S)[i], the neutral
  element  is  coloured in "light blue" and all other idempotents are coloured
  in "light coral"):
  
  3.3-3 DotForDrawingRightCayleyGraph
  
  DotForDrawingRightCayleyGraph( S )  function
  
  This  function  computes  the dot code that is used by the previous function
  and can also be used by the reader in some independent way.
  
  
  3.4 Schützenberger graphs
  
  3.4-1 DrawSchutzenbergerGraphs
  
  DrawSchutzenbergerGraphs( S )  function
  
  Draws the Schützenberger graphs of the inverse semigroup S.
  
  For example, DrawSchutzenbergerGraphs(poi3); would produce the following:
  
  
  3.5 Drawings output formats
  
  By  default,  when  a  drawing  is requested, it is outputted in pdf format.
  Since      graphviz      allows      other      output      formats     (see
  http://www.graphviz.org/doc/info/output.html),   it   is  possible  to  also
  request a drawing in a format other than pdf.
  
  3.5-1 DrawingsListOfExtraFormats
  
  DrawingsListOfExtraFormats global variable
  
  This  is  a  global  variable  which  holds  the  valid output formats for a
  drawing.  It  currently  has  the  value: ["dia", "fig", "gd", "gd2", "gif",
  "hpgl", "jpg", "mif", "mp", "pcl", "pic", "plain", "plain-ext", "png", "ps",
  "ps2",    "svg",    "svgz",    "vrml",    "vtx",    "wbmp",   "none"]   (see
  http://www.graphviz.org/doc/info/output.html for their description).
  
  3.5-2 DrawingsExtraFormat
  
  DrawingsExtraFormat global variable
  
  This  is  a  global variable which holds the alternative output format for a
  drawing.  By  default its value is "none" which indicates that just pdf will
  be used as the output format.
  
  If  its  value  becomes  one of those in DrawingsListOfExtraFormats (3.5-1),
  then besides the pdf file, it will also be created a file in the alternative
  format.
  
  To change this variable's value, please use SetDrawingsExtraFormat (3.5-3).
  
  3.5-3 SetDrawingsExtraFormat
  
  SetDrawingsExtraFormat( f )  function
  
  This function is used to set the value of DrawingsExtraFormat (3.5-2) to the
  format f which is one of DrawingsListOfExtraFormats (3.5-1).
  
    Example  
    gap> DrawingsExtraFormat;
    "none"
    gap> SetDrawingsExtraFormat("jpg");
    gap> DrawingsExtraFormat;
    "jpg"
    
    gap> DrawRightCayleyGraph(poi3);
    Displaying file: /tmp/tmp.tpJqvI/cayleygraph.dot.ps
    The extra output format file: /tmp/tmp.tpJqvI/cayleygraph.dot.jpg
    has also been created.
    
  
  
  
  3.6 Drawings extra graph attributes
  
  The  functions  described in this subsection are intended to give the user a
  finer  control  over  the  final  drawing.  They  allow  to define the graph
  attributes  described in http://graphviz.org/doc/info/attrs.html. (Note that
  only  graph  attributes  are  allowed  to  be  defined,  not  edge  nor node
  attributes are supported yet.)
  
  3.6-1 DrawingsExtraGraphAttributes
  
  DrawingsExtraGraphAttributes global variable
  
  This  is  a  global  variable  which  holds a list of strings, each of which
  defines  a dot graph attribute. This variable is not intended to be modified
  by  the  user  directly, but can be used to check which extra attributes are
  currently     defined.     To     set    the    attributes,    please    use
  SetDrawingsExtraGraphAttributes   (3.6-2).  If  DrawingsExtraGraphAttributes
  holds  the  value  "none"  then  the  default dot settings will be used. Use
  ClearDrawingsExtraGraphAttributes (3.6-3) to set it to "none".
  
  3.6-2 SetDrawingsExtraGraphAttributes
  
  SetDrawingsExtraGraphAttributes( L )  function
  
  This  is  the  function  to  define  the  drawing's  graph  attributes  (see
  http://graphviz.org/doc/info/attrs.html for a list and explanation of them).
  the  argument  L  is  a  list  of strings, each of which defines a dot graph
  attribute.  For example, if we wanted to define the graph size to be 7x9 (in
  inches), we would call SetDrawingsExtraGraphAttributes(["size=7,9"]);.
  
  If  we  also wanted to define the graph to be displayed in landscape mode we
  would  call SetDrawingsExtraGraphAttributes(["size=7,9", "rotate=90"]);. If,
  in  addition  we  wanted to define the background colour to be, for example,
  pink,    we    would    call    SetDrawingsExtraGraphAttributes(["size=7,9",
  "rotate=90", "bgcolor=pink"]);.
  
  After  defining  the attributes, any command that creates a drawing will use
  the  last  defined  attributes. To set them back to the defaults, please use
  ClearDrawingsExtraGraphAttributes (3.6-3).
  
    Example  
    gap> DrawingsExtraGraphAttributes;
    "none"
    
    
    gap> SetDrawingsExtraGraphAttributes(["size=7,9"]);
    
    gap> DrawingsExtraGraphAttributes;
    [ "size=7,9" ]
    
    
    gap> SetDrawingsExtraGraphAttributes(["size=7,9", "rotate=90"]);
    
    gap> DrawingsExtraGraphAttributes;
    [ "size=7,9", "rotate=90" ]
    
    
    gap> SetDrawingsExtraGraphAttributes(["size=7,9", "rotate=90", "bgcolor=pink"]);
    
    gap> DrawingsExtraGraphAttributes;
    [ "size=7,9", "rotate=90", "bgcolor=pink" ]
    
    
    gap> ClearDrawingsExtraGraphAttributes();
    
    gap> DrawingsExtraGraphAttributes;
    "none"
  
  
  3.6-3 ClearDrawingsExtraGraphAttributes
  
  ClearDrawingsExtraGraphAttributes(  )  function
  
  This function sets the graph drawing attributes back to dot's defaults.