3 Watch and Influence the Communication
  
  
  3.1 Functions
  
  3.1-1 homalgIOMode
  
  homalgIOMode( str[, str2[, str3]] )  function
  
  This  function  sets  different  modes  which  influence  how  much  of  the
  communication   becomes   visible.   Handling   the   string   str   is  not
  case-sensitive.  homalgIOMode invokes the global function homalgMode defined
  in  the  homalg  package  with  an  appropriate  argument  (see code below).
  Alternatively,  if  a  second  or more strings are given, then homalgMode is
  invoked  with  the  remaining  strings  str2,  str3,  ...  at  the  end.  In
  particular,  you  can  use  homalgIOMode(  str,  "" ) to reset the effect of
  invoking homalgMode.
  
        str │ str (long form) │ mode description                                                     
        ────┼─────────────────┼───────────────────────────────────────────────────────────────────  
            │                 │                                                                      
        ""  │       ""        │ the default mode, i.e. the communication protocol won't be visible   
            │                 │ (homalgIOMode( ) is a short form for homalgIOMode( "" ))             
            │                 │                                                                      
        "a" │      "all"      │ combine the modes "debug" and "file"                                 
            │                 │                                                                      
        "b" │     "basic"     │ the same as "picto" + homalgMode( "basic" )                          
            │                 │                                                                      
        "d" │     "debug"     │ view the complete communication protocol                             
            │                 │                                                                      
        "f" │     "file"      │ dump the communication protocol into a file with the name            
            │                 │ Concatenation( "commands_file_of_", CAS, "_with_PID_", PID )         
            │                 │                                                                      
        "p" │     "picto"     │ view the abbreviated communication protocol                          
            │                 │ using the preassigned pictograms                                     
            │                 │                                                                      
        ────┴─────────────────┴───────────────────────────────────────────────────────────────────  
  
  All  modes  other than the "default"-mode only set their specific values and
  leave  the  other  values  untouched,  which  allows  combining them to some
  extent.  This  also  means  that in order to get from one mode to a new mode
  (without  the  aim to combine them) one needs to reset to the "default"-mode
  first.
  Caution:
  
      In  case you choose one of the modes "file" or "all" you might want to
        set  the  global variable HOMALG_IO.DoNotDeleteTemporaryFiles := true;
        this  is  only  important if during the computations some matrices get
        converted  via  files  (using  ConvertHomalgMatrixViaFile), as reading
        these files will be part of the protocol!
  
      It   makes   sense   for  the  dumped  communication  protocol  to  be
        (re)executed  with  the  respective  external system, only in case the
        latter is deterministic (i.e. same-input-same-output).
  
    Code  
    InstallGlobalFunction( homalgIOMode,
      function( arg )
        local nargs, mode, s;
        
        nargs := Length( arg );
        
        if nargs = 0 or ( IsString( arg[1] ) and arg[1] = "" ) then
            mode := "default";
        elif IsString( arg[1] ) then	## now we know, the string is not empty
            s := arg[1];
            if LowercaseString( s{[1]} ) = "a" then
                mode := "all";
            elif LowercaseString( s{[1]} ) = "b" then
                mode := "basic";
            elif LowercaseString( s{[1]} ) = "d" then
                mode := "debug";
            elif LowercaseString( s{[1]} ) = "f" then
                mode := "file";
            elif LowercaseString( s{[1]} ) = "p" then
                mode := "picto";
            else
                mode := "";
            fi;
        else
            Error( "the first argument must be a string\n" );
        fi;
        
        if mode = "default" then
            ## reset to the default values
            HOMALG_IO.color_display := false;
            HOMALG_IO.show_banners := true;
            HOMALG_IO.save_CAS_commands_to_file := false;
            HOMALG_IO.DoNotDeleteTemporaryFiles := false;
            HOMALG_IO.SaveHomalgMaximumBackStream := false;
            HOMALG_IO.InformAboutCASystemsWithoutActiveRings := true;
            SetInfoLevel( InfoHomalgToCAS, 1 );
            homalgMode( );
        elif mode = "all" then
            homalgIOMode( "debug" );
            homalgIOMode( "file" );
        elif mode = "basic" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 4 );
            homalgMode( "basic" );	## use homalgIOMode( "basic", "" ) to reset
        elif mode = "debug" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 8 );
            homalgMode( "debug" );	## use homalgIOMode( "debug", "" ) to reset
        elif mode = "file" then
            HOMALG_IO.save_CAS_commands_to_file := true;
        elif mode = "picto" then
            HOMALG_IO.color_display := true;
            HOMALG_IO.show_banners := true;
            SetInfoLevel( InfoHomalgToCAS, 4 );
            homalgMode( "logic" );	## use homalgIOMode( "picto", "" ) to reset
        fi;
        
        if nargs > 1 and IsString( arg[2] ) then
            CallFuncList( homalgMode, arg{[ 2 .. nargs ]} );
        fi;
        
    end );
  
  
  This is the part of the global function homalgSendBlocking that controls the
  visibility of the communication.
  
    Code  
    io_info_level := InfoLevel( InfoHomalgToCAS );
    
    if not IsBound( pictogram ) then
        pictogram := HOMALG_IO.Pictograms.unknown;
        picto := pictogram;
    elif io_info_level >= 3 then
        picto := pictogram;
        ## add colors to the pictograms
        if pictogram = HOMALG_IO.Pictograms.ReducedEchelonForm and
           IsBound( HOMALG_MATRICES.color_BOE ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOE, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.BasisOfModule and
          IsBound( HOMALG_MATRICES.color_BOB ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOB, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.DecideZero and
          IsBound( HOMALG_MATRICES.color_BOD ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOD, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.SyzygiesGenerators and
          IsBound( HOMALG_MATRICES.color_BOH ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOH, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.BasisCoeff and
          IsBound( HOMALG_MATRICES.color_BOC ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOC, pictogram, "\033[0m" );
        elif pictogram = HOMALG_IO.Pictograms.DecideZeroEffectively and
          IsBound( HOMALG_MATRICES.color_BOP ) then
            pictogram := Concatenation( HOMALG_MATRICES.color_BOP, pictogram, "\033[0m" );
        elif need_output or need_display then
            pictogram := Concatenation( HOMALG_IO.Pictograms.color_need_output,
                                 pictogram, "\033[0m" );
        else
            pictogram := Concatenation( HOMALG_IO.Pictograms.color_need_command,
                                 pictogram, "\033[0m" );
        fi;
    else
        picto := pictogram;
    fi;
    
    if io_info_level >= 3 then
        if ( io_info_level >= 7 and not need_display ) or io_info_level >= 8 then
            ## print the pictogram, the prompt of the external system,
            ## and the sent command
            Info( InfoHomalgToCAS, 7, pictogram, " ", stream.prompt,
                  L{[ 1 .. Length( L ) - 1 ]} );
        elif io_info_level >= 4 then
            ## print the pictogram and the prompt of the external system
            Info( InfoHomalgToCAS, 4, pictogram, " ", stream.prompt, "..." );
        else
            ## print the pictogram only
            Info( InfoHomalgToCAS, 3, pictogram );
        fi;
        
    fi;
  
  
  
  3.2 The Pictograms
  
  3.2-1 HOMALG_IO.Pictograms
  
  HOMALG_IO.Pictograms global variable
  
  The record of pictograms is a component of the record HOMALG_IO.
  
    Code  
    Pictograms := rec(
      
      ##
      ## colors:
      ##
      
      ## pictogram color of a "need_command" or assignment operation:
      color_need_command                      := "\033[1;33;44m",
      
      ## pictogram color of a "need_output" or "need_display" operation:
      color_need_output                       := "\033[1;34;43m",
      
      ##
      ## good morning computer algebra system:
      ##
      
      ## initialize:
      initialize                              := "ini",
      
      ## define macros:
      define                                  := "def",
      
      ## get time:
      time                                    := ":ms",
      
      ## memory usage:
      memory                                  := "mem",
      
      ## unknown:
      unknown                                 := "???",
      
      ##
      ## external garbage collection:
      ##
      
      ## delete a variable:
      delete                                  := "xxx",
      
      ## delete serveral variables:
      multiple_delete                         := "XXX",
      
      ## trigger the garbage collector:
      garbage_collector                       := "grb",
      
      ##
      ## create lists:
      ##
      
      ## define a list:
      CreateList                              := "lst",
      
      ##
      ## create rings:
      ##
      
      ## define a ring:
      CreateHomalgRing                        := "R:=",
      
      ## get the names of the "variables" defining the ring:
      variables                               := "var",
      
      ## define zero:
      Zero                                    := "0:=",
      
      ## define one:
      One                                     := "1:=",
      
      ## define minus one:
      MinusOne                                := "-:=",
      
      ##
      ## mandatory ring operations:
      ##
      
      ## get the name of an element:
      ## (important if the CAS pretty-prints ring elements,
      ##  we need names that can be used as input!)
      ## (install a method instead of a homalgTable entry)
      homalgSetName                           := "\"a\"",
      
      ## a = 0 ?
      IsZero                                  := "a=0",
      
      ## a = 1 ?
      IsOne                                   := "a=1",
      
      ## substract two ring elements
      ## (needed by SimplerEquivalentMatrix in case
      ##  CopyRow/ColumnToIdentityMatrix are not defined):
      Minus                                   := "a-b",
      
      ## divide the element a by the unit u
      ## (needed by SimplerEquivalentMatrix in case
      ##  DivideEntryByUnit is not defined):
      DivideByUnit                            := "a/u",
      
      ## important ring operations:
      ## (important for performance since existing
      ##  fallback methods cause a lot of traffic):
      
      ## is u a unit?
      ## (mainly needed by the fallback methods for matrices, see below):
      IsUnit                                  := "?/u",
      
      ##
      ## optional ring operations:
      ##
      
      ## copy an element:
      CopyElement                             := "a>a",
      
      ## add two ring elements:
      Sum                                     := "a+b",
      
      ## multiply two ring elements:
      Product                                 := "a*b",
      
      ## the (greatest) common divisor:
      Gcd                                     := "gcd",
      
      ## cancel the (greatest) common divisor:
      CancelGcd                               := "ccd",
      
      ## random polynomial:
      RandomPol                               := "rpl",
      
      ## numerator:
      Numerator                               := "num",
      
      ## denominator:
      Denominator                             := "den",
      
      ## evaluate polynomial:
      Evaluate                                := "evl",
      
      ## degree of a multivariate polynomial
      DegreeOfRingElement                     := "deg",
      
      ## is irreducible:
      IsIrreducible                           := "irr",
      
      ##
      ## create matrices:
      ##
      
      ## define a matrix:
      HomalgMatrix                            := "A:=",
      
      ## copy a matrix:
      CopyMatrix                              := "A>A",
      
      ## load a matrix from file:
      LoadHomalgMatrixFromFile                := "A<<",
      
      ## save a matrix to file:
      SaveHomalgMatrixToFile                  := "A>>",
      
      ## get a matrix entry as a string:
      MatElm                  := "<ij",
      
      ## set a matrix entry from a string:
      SetMatElm                  := ">ij",
      
      ## add to a matrix entry from a string:
      AddToMatElm                := "+ij",
      
      ## get a list of the matrix entries as a string:
      GetListOfHomalgMatrixAsString           := "\"A\"",
      
      ## get a listlist of the matrix entries as a string:
      GetListListOfHomalgMatrixAsString       := "\"A\"",
      
      ## get a "sparse" list of the matrix entries as a string:
      GetSparseListOfHomalgMatrixAsString     := ".A.",
      
      ## assign a "sparse" list of matrix entries to a variable:
      sparse                                  := "spr",
      
      ## list of assumed inequalities:
      Inequalities                            := "<>0",
      
      ## list of assumed inequalities:
      MaximalIndependentSet                   := "idp",
      
      ##
      ## mandatory matrix operations:
      ##
      
      ## test if a matrix is the zero matrix:
      ## CAUTION: the external system must be able to check
      ##          if the matrix is zero modulo possible ring relations
      ##          only known to the external system!
      IsZeroMatrix                            := "A=0",
      
      ## number of rows:
      NrRows                                  := "#==",
      
      ## number of columns:
      NrColumns                               := "#||",
      
      ## determinant of a matrix over a (commutative) ring:
      Determinant                             := "det",
      
      ## create a zero matrix:
      ZeroMatrix                              := "(0)",
      
      ## create a initial zero matrix:
      InitialMatrix                           := "[0]",
      
      ## create an identity matrix:
      IdentityMatrix                          := "(1)",
      
      ## create an initial identity matrix:
      InitialIdentityMatrix                   := "[1]",
      
      ## "transpose" a matrix (with "the" involution of the ring):
      Involution                              := "A^*",
      
      ## get certain rows of a matrix:
      CertainRows                             := "===",
      
      ## get certain columns of a matrix:
      CertainColumns                          := "|||",
      
      ## stack to matrices vertically:
      UnionOfRows                             := "A_B",
      
      ## glue to matrices horizontally:
      UnionOfColumns                          := "A|B",
      
      ## create a block diagonal matrix:
      DiagMat                                 := "A\\B",
      
      ## the Kronecker (tensor) product of two matrices:
      KroneckerMat                            := "AoB",
      
      ## multiply a ring element with a matrix:
      MulMat                                  := "a*A",
      
      ## multiply a matrix with a ring element:
      MulMatRight                             := "A*a",
      
      ## add two matrices:
      AddMat                                  := "A+B",
      
      ## substract two matrices:
      SubMat                                  := "A-B",
      
      ## multiply two matrices:
      Compose                                 := "A*B",
      
      ## pullback a matrix by a ring map:
      Pullback                                := "pbk",
      
      ##
      ## important matrix operations:
      ## (important for performance since existing
      ##  fallback methods cause a lot of traffic):
      ##
      
      ## test if two matrices are equal:
      ## CAUTION: the external system must be able to check
      ##          equality of the two matrices modulo possible ring relations
      ##          only known to the external system!
      AreEqualMatrices                        := "A=B",
      
      ## test if a matrix is the identity matrix:
      IsIdentityMatrix                        := "A=1",
      
      ## test if a matrix is diagonal (needed by the display method):
      IsDiagonalMatrix                        := "A=\\",
      
      ## get the positions of the zero rows:
      ZeroRows                                := "0==",
      
      ## get the positions of the zero columns:
      ZeroColumns                             := "0||",
      
      ## get "column-independent" unit positions
      ## (needed by ReducedBasisOfModule):
      GetColumnIndependentUnitPositions       := "ciu",
      
      ## get "row-independent" unit positions
      ## (needed by ReducedBasisOfModule):
      GetRowIndependentUnitPositions          := "riu",
      
      ## get the position of the "first" unit in the matrix
      ## (needed by SimplerEquivalentMatrix):
      GetUnitPosition                         := "gup",
      
      ## position of the first non-zero entry per row
      PositionOfFirstNonZeroEntryPerRow       := "fnr",
      
      ## position of the first non-zero entry per column
      PositionOfFirstNonZeroEntryPerColumn    := "fnc",
      
      ## indicator matrix of non-zero entries
      IndicatorMatrixOfNonZeroEntries         := "<>0",
      
      ## transposed matrix:
      TransposedMatrix                        := "^tr",
      
      ## divide an entry of a matrix by a unit
      ## (needed by SimplerEquivalentMatrix in case
      ##  DivideRow/ColumnByUnit are not defined):
      DivideEntryByUnit                       := "ij/",
      
      ## divide a row by a unit
      ## (needed by SimplerEquivalentMatrix):
      DivideRowByUnit                         := "-/u",
      
      ## divide a column by a unit
      ## (needed by SimplerEquivalentMatrix):
      DivideColumnByUnit                      := "|/u",
      
      ## divide a row by a unit
      ## (needed by SimplerEquivalentMatrix):
      CopyRowToIdentityMatrix                 := "->-",
      
      ## divide a column by a unit
      ## (needed by SimplerEquivalentMatrix):
      CopyColumnToIdentityMatrix              := "|>|",
      
      ## set a column (except a certain row) to zero
      ## (needed by SimplerEquivalentMatrix):
      SetColumnToZero                         := "|=0",
      
      ## get the positions of the rows with a single one
      ## (needed by SimplerEquivalentMatrix):
      GetCleanRowsPositions                   := "crp",
      
      ## convert a single row matrix into a matrix
      ## with specified number of rows/columns
      ## (needed by the display methods for homomorphisms):
      ConvertRowToMatrix                      := "-%A",
      
      ## convert a single column matrix into a matrix
      ## with specified number of rows/columns
      ## (needed by the display methods for homomorphisms):
      ConvertColumnToMatrix                   := "|%A",
      
      ## convert a matrix into a single row matrix:
      ConvertMatrixToRow                      := "A%-",
      
      ## convert a matrix into a single column matrix:
      ConvertMatrixToColumn                   := "A%|",
      
      ##
      ## basic matrix operations:
      ##
      
      ## compute a (r)educed (e)chelon (f)orm:
      ReducedEchelonForm                      := "ref",
      
      ## compute a "(bas)is" of a given set of module elements:
      BasisOfModule                           := "bas",
      
      ## compute a reduced "(Bas)is" of a given set of module elements:
      ReducedBasisOfModule                    := "Bas",
      
      ## (d)e(c)ide the ideal/submodule membership problem,
      ## i.e. if an element is (0) modulo the ideal/submodule:
      DecideZero                              := "dc0",
      
      ## compute a generating set of (syz)ygies:
      SyzygiesGenerators                      := "syz",
      
      ## compute a generating set of reduced (Syz)ygies:
      ReducedSyzygiesGenerators               := "Syz",
      
      ## compute a (R)educed (E)chelon (F)orm
      ## together with the matrix of coefficients:
      ReducedEchelonFormC                     := "REF",
      
      ## compute a "(BAS)is" of a given set of module elements
      ## together with the matrix of coefficients:
      BasisCoeff                              := "BAS",
      
      ## (D)e(C)ide the ideal/submodule membership problem,
      ## i.e. write an element effectively as (0) modulo the ideal/submodule:
      DecideZeroEffectively                   := "DC0",
      
      ##
      ## optional matrix operations:
      ##
      
      ## Hilbert-Poincare series of a module:
      HilbertPoincareSeries                   := "HPs",
      
      ## Hilbert polynomial of a module:
      HilbertPolynomial                       := "Hil",
      
      ## affine dimension of a module:
      AffineDimension                         := "dim",
      
      ## affine degree of a module:
      AffineDegree                            := "adg",
      
      ## the constant term of the hilbert polynomial:
      ConstantTermOfHilbertPolynomial         := "P_0",
      
      ## primary decomposition:
      PrimaryDecomposition                    := "YxZ",
      
      ## eliminate variables:
      Eliminate                               := "eli",
      
      ## leading module:
      LeadingModule                           := "led",
      
      ## the i-th monomial matrix
      MonomialMatrix                          := "mon",
      
      ## matrix of symbols:
      MatrixOfSymbols                         := "smb",
      
      ## leading module:
      ## coefficients:
      Coefficients                            := "cfs",
      
      ##
      ## optional module operations:
      ##
      
      ## compute a better equivalent matrix
      ## (field -> row+col Gauss, PIR -> Smith, Dedekind domain -> Krull, etc ...):
      BestBasis                               := "(\\)",
      
      ## compute elementary divisors:
      ElementaryDivisors                      := "div",
      
      ##
      ## for the eye:
      ##
      
      ## display objects:
      Display                                 := "dsp",
      
      ## the LaTeX code of the mathematical entity:
      homalgLaTeX                             := "TeX",
      
    )