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

#############################################################################
##
#W  ringhom.gi                   GAP library                  Alexander Hulpke
##
##
#Y  Copyright (C) 2008 The GAP Group
##
##  This file contains methods for ring general mappings and homomorphisms.
##  It is based on alghom.gi
##


#############################################################################
##
#R  IsRingGeneralMappingByImagesDefaultRep
#R  IsSCRingGeneralMappingByImagesDefaultRep
##
##
DeclareRepresentation( "IsRingGeneralMappingByImagesDefaultRep",
    IsRingGeneralMapping and IsAdditiveElementWithInverse
    and IsAttributeStoringRep, [] );

DeclareRepresentation( "IsSCRingGeneralMappingByImagesDefaultRep",
    IsRingGeneralMappingByImagesDefaultRep,[]);


#############################################################################
##
#M  RingGeneralMappingByImages( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( RingGeneralMappingByImages,
    "for two rings and two homogeneous lists",
    [ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ],
function( S, R, gens, imgs )
  local filter,map;        # general mapping from <S> to <R>, result

  # Check the arguments.
  if   Length( gens ) <> Length( imgs )  then
    Error( "<gens> and <imgs> must have the same length" );
  elif not IsSubset( S, gens ) then
    Error( "<gens> must lie in <S>" );
  elif not IsSubset( R, imgs ) then
    Error( "<imgs> must lie in <R>" );
  fi;
  filter:=IsSPGeneralMapping and IsRingGeneralMapping;

  if IsSubringSCRing(S) then
    filter:=filter and IsSCRingGeneralMappingByImagesDefaultRep;
  fi;

  # Make the general mapping.
  map:= Objectify( TypeOfDefaultGeneralMapping( S, R,
			    IsSPGeneralMapping
			and IsRingGeneralMapping
			and IsSCRingGeneralMappingByImagesDefaultRep ),
		    rec(
			) );

    SetMappingGeneratorsImages(map,[Immutable(gens),Immutable(imgs)]);
    # return the general mapping
    return map;
    end );

#############################################################################
##
#M  RingHomomorphismByImagesNC( <S>, <R>, <gens>, <imgs> )
##
InstallMethod( RingHomomorphismByImagesNC,
    "for two rings and two homogeneous lists",
    [ IsRing, IsRing, IsHomogeneousList, IsHomogeneousList ],
    function( S, R, gens, imgs )
    local map;        # homomorphism from <source> to <range>, result
    map:= RingGeneralMappingByImages( S, R, gens, imgs );
    SetIsSingleValued( map, true );
    SetIsTotal( map, true );
    return map;
    end );


#############################################################################
##
#F  RingHomomorphismByImages( <S>, <R>, <gens>, <imgs> )
##
InstallGlobalFunction( RingHomomorphismByImages,
    function( S, R, gens, imgs )
    local hom;
    hom:= RingGeneralMappingByImages( S, R, gens, imgs );
    if IsMapping( hom ) then
      return RingHomomorphismByImagesNC( S, R, gens, imgs );
    else
      return fail;
    fi;
end );

#############################################################################
##
#M  ViewObj( <map> )  . . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ViewObj, "for an ring g.m.b.i", true,
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  View(mapi[1]);
  Print(" -> ");
  View(mapi[2]);
end );


#############################################################################
##
#M  PrintObj( <map> ) . . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( PrintObj, "for an ring hom. b.i.", true,
    [     IsMapping
      and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  Print( "RingHomomorphismByImages( ",
	  Source( map ), ", ", Range( map ), ", ",
	  mapi[1], ", ", mapi[2], " )" );
end );

InstallMethod( PrintObj, "for an ring g.m.b.i", true,
    [     IsGeneralMapping
      and IsRingGeneralMappingByImagesDefaultRep ], 0,
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  Print( "RingGeneralMappingByImages( ",
	  Source( map ), ", ", Range( map ), ", ",
	  mapi[1], ", ", mapi[2], " )" );
end );

#############################################################################
##
#M  IsTotal( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsTotal,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local mapi,t;
  mapi:=MappingGeneratorsImages(map);
  if Length(mapi[1])=0 then
    t:=Ring(Zero(Source(map)));
  else
    t:=Ring(mapi[1]);
  fi;
  return Source(map)=t;
end);

#############################################################################
##
#M  MakeSCRingMapping( <map> )
##
BindGlobal( "MakeSCRingMapping",
function(map)
local mapi;
  if not IsBound(map!.stdgens) then
    mapi:=MappingGeneratorsImages(map);
    map!.stdgens:=
      StandardGeneratorsImagesSubringSCRing(FamilyObj(Zero(Source(map))),
	mapi[1],mapi[2]);
  fi;
end);

#############################################################################
##
#M  IsSingleValued( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsSingleValued,
    "for sc ring g.m.b.i.",
    [ IsGeneralMapping and IsSCRingGeneralMappingByImagesDefaultRep ],
function(map)
  local r, moduli, std, stdi, sel, o, elm, elmi, i, j, k;
  r:=Source(map);
  moduli:=FamilyObj(Zero(r))!.moduli;

  MakeSCRingMapping(map);
  std:=map!.stdgens;
  # check additive relations
  for i in [1..Length(std[4])] do
    stdi:=std[1][i];
    sel:=Filtered([1..Length(stdi)],x->stdi[x]<>0);
    if not 0 in moduli{sel} then
      o:=1;
      for j in sel do
        o:=Lcm(o,moduli[j]/Gcd(stdi[j],moduli[j]));
      od;
      if not IsZero(o*std[4][i]) then
	Info(InfoRingHom,2,"Additive order ",o," of generator ",i," failed");
	return false;
      else
	Info(InfoRingHom,3,"Additive order ",o," of generator ",i," OK");
      fi;
    else
      Info(InfoRingHom,3,"Generator ",i,": ",std[1][i]," has order infinity");
    fi;
  od;

  # check multiplicative relations
  for i in [1..Length(std[4])] do
    for j in [1..Length(std[4])] do
      elm:=std[3][i]*std[3][j];
      elm:=SCRingDecompositionStandardGens(std,elm);
      elmi:=Zero(Range(map));
      for k in [1..Length(std[4])] do
        elmi:=elmi+elm[k]*std[4][k];
      od;
      if elmi<>std[4][i]*std[4][j] then
	Info(InfoRingHom,2,"Product ",i," x ",j," failed: ",elm,elmi);
	return false;
      fi;
    od;
  od;
  return true;
end);


#############################################################################
##
#M  ImagesSource( <map> ) . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ImagesSource,
    "for an ring g.m.b.i.",
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  return Subring(Range(map),MappingGeneratorsImages(map)[2]);
end );

#############################################################################
##
#M  PreImagesRange( <map> ) . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( PreImagesRange,
    "for an ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  return Subring(Source(map),MappingGeneratorsImages(map)[1]);
end );

#############################################################################
##
#M  InverseGeneralMapping( <map> ) . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( InverseGeneralMapping,
    "for an ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
local mapi;
  mapi:=MappingGeneratorsImages(map);
  return RingGeneralMappingByImages(Range(map),Source(map),mapi[2],mapi[1]);
end );

#############################################################################
##
#M  AdditiveInverseOp( <map> )  . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( AdditiveInverseOp, "for ring g.m.b.i.",
  [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local mapi;
  mapi:=MappingGeneratorsImages(map);
  return RingGeneralMappingByImages(Source(map),Range(map),mapi[1],
    List(mapi[2],AdditiveInverse));
end);

#############################################################################
##
#M  \+( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \+,
    "for ring g.m.b.i. and ring general mapping",
    IsIdenticalObj,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsRingGeneralMapping],
function( map1, map2 )
local mapi,map;
  mapi:=MappingGeneratorsImages(map1);
  map:=RingGeneralMappingByImages(Source(map1),Range(map1),mapi[1],
    List([1..Length(mapi[1])],
	  x->mapi[2][x]+ImagesRepresentative(map2,mapi[1][x])));
  return map;
end );

#############################################################################
##
#M  \+( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \+,
    "for ring ring general mapping and ring g.m.b.i.",
    IsIdenticalObj,
    [ IsRingGeneralMapping,
      IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep],
function( map2, map1 )
local mapi,map;
  mapi:=MappingGeneratorsImages(map1);
  map:=RingGeneralMappingByImages(Source(map1),Range(map1),mapi[1],
    List([1..Length(mapi[1])],
	  x->mapi[2][x]+ImagesRepresentative(map2,mapi[1][x])));
  return map;
end );

#############################################################################
##
#M  \=( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \=,
    "for ring g.m.b.i. and ring general mapping",
    IsIdenticalObj,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsRingGeneralMapping],
function( map1, map2 )
local mapi;
  mapi:=MappingGeneratorsImages(map1);
  return ForAll([1..Length(mapi[1])],
	  x->mapi[2][x]=ImagesRepresentative(map2,mapi[1][x]));
end );

#############################################################################
##
#M  \=( <map1>, map2> ) . . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
##
InstallOtherMethod( \=,
    "for ring ring general mapping and ring g.m.b.i.",
    IsIdenticalObj,
    [ IsRingGeneralMapping,
      IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep],
function( map2, map1 )
local mapi;
  mapi:=MappingGeneratorsImages(map1);
  return ForAll([1..Length(mapi[1])],
	  x->mapi[2][x]=ImagesRepresentative(map2,mapi[1][x]));
end );

#############################################################################
##
#M  CoKernelOfAdditiveGeneralMapping( <map> ) . . . . .  for ring g.m.b.i.
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
local r, moduli, std, gens, stdi, sel, o, elm, elmi, i, j, k;
  r:=Source(map);
  moduli:=FamilyObj(Zero(r))!.moduli;

  MakeSCRingMapping(map);
  std:=map!.stdgens;
  gens:=[];
  # run through additive relations
  for i in [1..Length(std[4])] do
    stdi:=std[1][i];
    sel:=Filtered([1..Length(stdi)],x->stdi[x]<>0);
    if not 0 in moduli{sel} then
      o:=1;
      for j in sel do
        o:=Lcm(o,moduli[j]/Gcd(stdi[j],moduli[j]));
      od;
      Add(gens,o*std[4][i]);
    fi;
  od;

  # check multiplicative relations
  for i in [1..Length(std[4])] do
    for j in [1..Length(std[4])] do
      elm:=std[3][i]*std[3][j];
      elm:=SCRingDecompositionStandardGens(std,elm);
      elmi:=Zero(Range(map));
      for k in [1..Length(std[4])] do
        elmi:=elmi+elm[k]*std[4][k];
      od;
      Add(gens,elmi-std[4][i]*std[4][j]);
    od;
  od;
  gens:=Filtered(gens,i->not IsZero(i));
  if Length(gens)=0 then Add(gens,Zero(Range(map)));fi;
  return Subring(Range(map),gens);
end);



#############################################################################
##
#M  KernelOfAdditiveGeneralMapping( <map> ) . . . . . .  for ring g.m.b.i.
##
InstallMethod( KernelOfAdditiveGeneralMapping,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function( map )
  local ker, mapi, i;
  ker:=ShallowCopy(GeneratorsOfRing(
      CoKernelOfAdditiveGeneralMapping(InverseGeneralMapping(map))));
  mapi:=MappingGeneratorsImages(map);
  for i in [1..Length(mapi[1])] do
    if IsZero(mapi[2][i]) then
      Add(ker,mapi[1][i]);
    fi;
  od;
  return Subring(Source(map),ker);
end );


#############################################################################
##
#M  IsInjective( <map> )  . . . . . . . . . . . . . . .  for ring g.m.b.i.
##
InstallMethod( IsInjective,
    "for ring g.m.b.i.",
    [ IsGeneralMapping and IsRingGeneralMappingByImagesDefaultRep ],
function(map)
  return Size(KernelOfAdditiveGeneralMapping(map))=1;
end);


#############################################################################
##
#M  ImagesRepresentative( <map>, <elm> )  . . . . . . .  for ring g.m.b.i.
##
InstallMethod( ImagesRepresentative, "for SC ring g.m.b.i., and element",
    FamSourceEqFamElm,
    [ IsRingGeneralMapping and IsSCRingGeneralMappingByImagesDefaultRep,
      IsObject ],
function( map, elm )
local std, elmi, k;
  MakeSCRingMapping(map);
  std:=map!.stdgens;
  elm:=SCRingDecompositionStandardGens(std,elm);
  elmi:=Zero(Range(map));
  for k in [1..Length(std[4])] do
    elmi:=elmi+elm[k]*std[4][k];
  od;
  return elmi;
end );

#############################################################################
##
#M  PreImagesRepresentative( <map>, <elm> ) . . . . . .  for ring g.m.b.i.
##
InstallMethod( PreImagesRepresentative,
    "for ring g.m.b.i., and element",
    FamRangeEqFamElm,
    [ IsRingGeneralMapping and IsRingGeneralMappingByImagesDefaultRep,
      IsObject ],
function( map, elm )
  return ImagesRepresentative(InverseGeneralMapping(map),elm);
end );

BindGlobal("IsomorphismSCRing",function(R)
local e, z, one, o, sel, g, go, elms, dec, p, cands, m, a, b, nr, hom, i, j;
  if Size(R)>100000 then
    Error("R is too big");
  fi;
  # find generators
  e:=Elements(R);
  z:=Zero(R);
  one:=One(R);
  one:=Position(e,one);
  o:=List(e,i->First([1..Size(R)],x->x*i=z));
  sel:=[1..Length(e)];
  g:=[];
  go:=[];
  elms:=[z];
  dec:=[];
  p:=Position(e,z);
  dec[p]:=[];
  RemoveSet(sel,p);
  cands:=ShallowCopy(sel);
  while Length(cands)>0 do

    # element of maximal order. If possible pick ``one'' to be among the
    # generators
    m:=Maximum(o{cands});
    if one in cands and o[one]=m then
      a:=one;
    else
      a:=First(cands,i->o[i]=m);
    fi;
    RemoveSet(cands,a);
    a:=e[a];
    Add(g,a);
    Add(go,m);
    # all combinations
    for i in [1..Length(elms)] do
      for j in [1..m-1] do
	b:=elms[i]+j*a;
	p:=Position(e,b);
	if p in sel then
	  RemoveSet(sel,p);
	  Add(elms,b);
	  Add(dec,Concatenation(dec[i],[j]));
	fi;
      od;
    od;

    # the remaining candidates must be complements
    for i in ShallowCopy(cands) do
      if ForAny([1..o[i]-1],j->j*e[i] in elms) then
	RemoveSet(cands,i);
      fi;
    od;

    # update dec
    m:=Length(g);
    for i in dec do
      while Length(i)<m do
	Add(i,0);
      od;
    od;
  od;
  m:=EmptySCTable(Length(go),0);
  for i in [1..Length(g)] do
    for j in [1..Length(g)] do
      p:=g[i]*g[j];
      if p<>z then
	p:=Position(elms,p);
	p:=dec[p];
	nr:=[];
	for b in [1..Length(p)] do
	  if p[b]<>0 then
	    Add(nr,p[b]);
	    Add(nr,b);
	  fi;
	od;
	SetEntrySCTable(m,i,j,nr);
      fi;
    od;
  od;
  nr:=RingByStructureConstants(go,m);
  hom:=RingHomomorphismByImages(R,nr,g,GeneratorsOfRing(nr));
  return hom;
end);

#############################################################################
##
#M  NaturalHomomorphismByIdeal(<R>,<I>)
##
InstallMethod( NaturalHomomorphismByIdeal,"sc rings",IsIdenticalObj,
    [ IsSubringSCRing,IsSubringSCRing],
function( R, I )
  local hom, R2, nat, Rgens, std, moduli, newmod, posi, q, t, dec, x, i, j, k;
  if not IsIdeal(R,I) then
    Error("I is not an ideal!");
  fi;
  if not IsWholeFamily(R) then
    hom:=IsomorphismSCRing(R);
    R2:=Range(hom);
    I:=Subring(R2,List(GeneratorsOfRing(I),x->Image(hom,x)));
    nat:=NaturalHomomorphismByIdeal(R2,I);
    return RingHomomorphismByImages(R,Range(nat),GeneratorsOfRing(R),
             List(GeneratorsOfRing(R),x->Image(nat,Image(hom,i))));
  else
    if I=R then
      # catch trivial case
      q:=SmallRing(1,1);
      return RingHomomorphismByImages(R,q,GeneratorsOfRing(R),
        List(GeneratorsOfRing(R),x->Zero(q)));
    fi;
    # R is the full ring. We can read of the factor ring structures from the
    # standard generators of R
    Rgens:=GeneratorsOfRing(R);
    std:=StandardGeneratorsSubringSCRing(I);
    moduli:=FamilyObj(Zero(R))!.moduli;
    newmod:=[];
    posi:=[]; # generator positions
    for i in [1..Length(moduli)] do
      if not IsBound(std[2][i]) then
        # the generator survives as it is
	Add(newmod,moduli[i]);
	Add(posi,i);
      else
        t:=std[1][std[2][i]][i]; 
        if t=1 then
	  # the generator vanishes in the factor
	  Add(newmod,false);
	else
	  # the generator has a smaller order in the factor
	  q:=moduli[i]/t; 
	  Add(newmod,q);
	  Add(posi,i);
	fi;
      fi;
    od;
    # now determine the multiplication
    t:=EmptySCTable(Length(posi),0);
    for i in [1..Length(posi)] do
      for j in [1..Length(posi)] do
	# product of generators
        q:=Rgens[posi[i]]*Rgens[posi[j]];
	q:=q![1]; # the coefficients
	dec:=[];
	for k in [1..Length(posi)] do
	  x:=q[posi[k]] mod newmod[posi[k]];
	  if x<>0 then
	    Add(dec,x);
	    Add(dec,k);
	  fi;
	od;
	if Length(dec)>0 then
	  SetEntrySCTable(t,i,j,dec);
	fi;
      od;
    od;
  fi;
  q:=RingByStructureConstants(newmod{posi},t,"q");
  # image list: Generators are mapped to their images
  x:=List(GeneratorsOfRing(R),x->Zero(q));
  x{posi}:=GeneratorsOfRing(q);
  hom:=RingHomomorphismByImages(R,q,GeneratorsOfRing(R),x);
  SetIsSurjective(hom,true);
  SetKernelOfAdditiveGeneralMapping(hom,I);
  return hom;
end );

InstallOtherMethod( \/,
    "generic method for two rings",
    IsIdenticalObj,
    [ IsRing, IsRing ],
    function( R, I )
    return ImagesSource( NaturalHomomorphismByIdeal( R, I ) );
    end );