gap-libs 4r7p5-2 / usr / share / gap / lib / mgmadj.gi

#############################################################################
##
#W  mgmadj.gi                    GAP library                  Andrew Solomon
##
##
#Y  Copyright (C)  1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,  Germany
#Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y  Copyright (C) 2002 The GAP Group
##
##  This file contains generic methods for magmas with zero adjoined.
##

InstallMethod( IsMultiplicativeZero,
"for magma with multiplicative zero and multiplicative element",
IsCollsElms,
[ IsMagma and HasMultiplicativeZero, IsMultiplicativeElement],
function( M, z )
  return z = MultiplicativeZero(M);
end);

#

InstallMethod( IsMultiplicativeZero,
"for a magma and multiplicative element",
IsCollsElms,
[ IsMagma, IsMultiplicativeElement],
function(M, z)
  local i, en, x;

  i := 1;
  en := Enumerator(M);
  while IsBound(en[i]) do
    x := en[i];
    if x*z <> z or z*x <> z then	
      return false;
    fi;
    i := i +1;
  od;
  SetMultiplicativeZero(M,Immutable(z));
  return true;
end);

#

InstallMethod( IsMultiplicativeZero,
"for a semigroup with generators and multiplicative element",
IsCollsElms, 
[IsSemigroup and HasGeneratorsOfSemigroup, IsMultiplicativeElement],
function(S, z)
  if HasMultiplicativeZero(S) then 
    return z=MultiplicativeZero(S);
  elif ForAll(GeneratorsOfSemigroup(S), x->x*z=z and z*x=z) then
    SetMultiplicativeZero(S, Immutable(z));
    return true;
  fi;
  return false;
end);

#

InstallOtherMethod( MultiplicativeZero, "for a magma",
[ IsMagma ],
function( M )
  local en, i;

  en := Enumerator(M);
  i := 1;
  while (IsBound(en[i])) do
    if IsMultiplicativeZero(M, en[i]) then
      return en[i];
    fi;
    i := i +1;
  od;
  return fail;
end );

# MagmaWithZeroAdjoined

InstallMethod(MagmaWithZeroAdjoined, "for a magma with 0 adjoined element",
[IsMagmaWithZeroAdjoinedElementRep],
function( elm )
  return FamilyObj(elm)!.MagmaWithZeroAdjoined;
end);

#

InstallMethod(MagmaWithZeroAdjoined, "for a magma", 
[IsMagma], m-> Range(InjectionZeroMagma(m)));

#

InstallMethod( OneMutable, "for an element of a magma with zero adjoined",
[IsMultiplicativeElementWithOne and IsMagmaWithZeroAdjoinedElementRep],
x-> One(MagmaWithZeroAdjoined(x)));

#

InstallMethod( MultiplicativeZeroOp,
"for an element of a magma with zero adjoined",
[ IsMagmaWithZeroAdjoinedElementRep], 
function( elm )
  return MultiplicativeZero(MagmaWithZeroAdjoined(elm));  
end );

#

InstallMethod(InjectionZeroMagma, "for a magma",
[IsMagma],
function(m)
  local filts, fam, type, inj, zero, gens, out;
  
  if Length(GeneratorsOfMagma(m))=0 then
    Error("usage: it is only possible to adjoin a zero to a magma",  
    " with generators,");
    return;
  fi;
 
  # filters for the elements
  filts := IsMultiplicativeElementWithZero;

  if IsMultiplicativeElementWithOne(Representative(m)) then
    filts := filts and IsMultiplicativeElementWithOne;
  fi;

  if IsAssociativeElement(Representative(m)) then
    filts := filts and IsAssociativeElement;
  fi;

  fam:=NewFamily( "FamilyOfElementOfMagmaWithZeroAdjoined", filts);
  type:=NewType(fam, filts and IsMagmaWithZeroAdjoinedElementRep);
  
  #the injection
  inj:=function(elt)
    local new;
    new:=Objectify(type, rec(elt:=elt));
    return new;
  end;
  
  # set the one
  if IsMagmaWithOne(m) then
    SetOne(fam, inj(One(m)));
  fi;
 
  #filters for the magma with 0 adjoined
  filts:=IsAttributeStoringRep and IsMagmaWithZeroAdjoined;

  if IsSemigroup(m) then
    filts:=filts and IsAssociative;
  fi;

  if IsMagmaWithOne(m) then 
    filts:=filts and IsMagmaWithOne;
  fi;

  zero := Objectify(type, rec(elt:=fail));;
  gens:=Concatenation(List(GeneratorsOfMagma(m), g-> inj(g)), [zero]);
  out:=Objectify( NewType( FamilyObj( gens ), filts), rec());

  # store the magma in the family so that it can be recovered from an element
  fam!.MagmaWithZeroAdjoined:=out;

  if IsGroup(m) then
    SetIsZeroGroup(out, true);
  fi;

  SetMultiplicativeZero(out, zero); 
  
  if IsMagmaWithOne(out) then 
    SetGeneratorsOfMagmaWithOne(out, gens);
  fi;
  SetGeneratorsOfMagma(out, gens);

  inj:=MappingByFunction(m, out, inj, x-> x!.elt);
  SetUnderlyingInjectionZeroMagma(out, inj);
  
  return inj;
end);

#

InstallMethod(PrintObj, "for a magma with zero adjoined",
[IsMagmaWithZeroAdjoined and IsMagma and HasGeneratorsOfMagma], 10 ,
function(m) 
  Print("<");
  PrintObj(Source(UnderlyingInjectionZeroMagma(m)));
  Print(" with 0 adjoined>");
  return;
end);  

#

InstallMethod(ViewObj, "for a zero group",
[IsMagmaWithZeroAdjoined and IsMagma and HasGeneratorsOfMagma], 10, 
function(m)  
  Print("<");
  ViewObj(Source(UnderlyingInjectionZeroMagma(m)));
  Print(" with 0 adjoined>");
  return; 
end);

#

InstallMethod( Size, "for a magma with a zero adjoined",
[IsMagmaWithZeroAdjoined], 
function(m)
  return Size(Source(UnderlyingInjectionZeroMagma(m)))+1;
end);

#

InstallMethod( \*, "for two elements of a magma with zero adjoined",
IsIdenticalObj,
[ IsMagmaWithZeroAdjoinedElementRep, IsMagmaWithZeroAdjoinedElementRep ],
function(x, y)

  if x!.elt=fail then 
    return x;
  elif y!.elt=fail then	
    return y;
  fi;
  return (x!.elt*y!.elt)^UnderlyingInjectionZeroMagma(
   MagmaWithZeroAdjoined(x)); 
end );

#

InstallMethod( \=, "for two elements of a magma with zero adjoined",
IsIdenticalObj,
[IsMagmaWithZeroAdjoinedElementRep, IsMagmaWithZeroAdjoinedElementRep ],
function(x, y)
  return x!.elt=y!.elt;
end);

#  ordering of the underlying magma with zero less than everything else

InstallMethod(\<, "for two elements of magmas with zero adjoined", 
IsIdenticalObj,
[ IsMagmaWithZeroAdjoinedElementRep, IsMagmaWithZeroAdjoinedElementRep ],
function(x, y)
  local xx, yy;
 
  xx:=x!.elt; yy:=y!.elt;
    
  if xx=fail then 
    return not yy=fail;
  elif yy=fail then 
    return false;
  fi;
  return xx<yy;
end);

#

InstallMethod(PrintObj, "for an element of a magma with zero adjoined", 
[IsMagmaWithZeroAdjoinedElementRep],
function(x) 
  local m;
 
  m:=FamilyObj(x)!.MagmaWithZeroAdjoined;
  Print("<");
  if IsGroup(Source(UnderlyingInjectionZeroMagma(m))) then 
    Print("group ");
  elif IsMonoid(Source(UnderlyingInjectionZeroMagma(m))) then
    Print("monoid ");
  elif IsSemigroup(Source(UnderlyingInjectionZeroMagma(m))) then 
    Print("semigroup ");
  else 
    Print("magma ");
  fi;

  Print("with 0 adjoined elt: ");
  if x!.elt=fail then 
    Print("0");
  else 
    PrintObj(x!.elt);
  fi;
  Print(">");
  return;
end);