Quelle nrconstr.gi Sprache: unbekannt

## N!.multiplication ersetzt PM

#############################################################################
##
#M  DirectProductNearRing
##

InstallMethod(
DirectProductNearRing,
"ExpMulNR*ExpMulNR",
true,
[IsExplicitMultiplicationNearRing, IsExplicitMultiplicationNearRing],
0,
  function ( N, M )
  local NM, pr1, pr2, e1, e2, mul, D;
    NM := DirectProduct(GroupReduct(N),GroupReduct(M));
    e1 := Embedding( NM, 1 );
    e2 := Embedding( NM, 2 );
    pr1 := Projection( NM, 1 );
    pr2 := Projection( NM, 2 );
    mul := function( x, y )
      return Image(e1,NRMultiplication(N)(Image(pr1,x),Image(pr1,y)))
            *Image(e2,NRMultiplication(M)(Image(pr2,x),Image(pr2,y)));
    end;

    D := ExplicitMultiplicationNearRingNC( NM, mul );
    SetDirectProductNearRingFlag( D, true );
    # remember the factors
    D!.factors := [ N, M ];

    return D;
end );

#############################################################################
##
#M  DirectProductNearRing for 2 transformation nearrings
##

InstallMethod(
DirectProductNearRing,
"TfmNRs",
true,
[IsTransformationNearRing, IsTransformationNearRing],
0,
  function ( nr1, nr2 )
  local gamma1, gamma2, newGamma,
elms, fam,
pr1, pr2, emb1, emb2,
embedTransformation,
zero1, zero2,
addGens1, addGens2, addGens,
D;

    gamma1 := Gamma( nr1 ); gamma2 := Gamma( nr2 );

    newGamma := DirectProduct( gamma1, gamma2 );
    elms := AsSSortedList( newGamma );
    fam := EndoMappingFamily( newGamma );

    pr1 := Projection( newGamma, 1 );
    pr2 := Projection( newGamma, 2 );

    emb1 := Embedding( newGamma, 1 );
    emb2 := Embedding( newGamma, 2 );

    embedTransformation := function ( f, g )
    local vec, elm;
      vec := [];
      for elm in elms do
Add( vec, Position( elms,
  Image( emb1, Image( f, Image( pr1, elm ) ) ) *
  Image( emb2, Image( g, Image( pr2, elm ) ) )
    )              );
      od;
      return EndoMappingByTransformation( newGamma, fam, Transformation(vec) );
    end;

    zero1 := ConstantEndoMapping( gamma1, Identity( gamma1 ) );
    zero2 := ConstantEndoMapping( gamma2, Identity( gamma2 ) );

    addGens1 := List( GeneratorsOfGroup( GroupReduct( nr1 ) ),
   gen -> GroupGeneralMappingByGroupElement( GroupReduct(nr1), gen ) );
    addGens2 := List( GeneratorsOfGroup( GroupReduct( nr2 ) ),
   gen -> GroupGeneralMappingByGroupElement( GroupReduct(nr2), gen ) );

    addGens := Concatenation(
  List( addGens1, gen -> embedTransformation( gen, zero2 ) ),
  List( addGens2, gen -> embedTransformation( zero1, gen ) ) );

    D := TransformationNearRingByAdditiveGenerators( newGamma, addGens );

    SetDirectProductNearRingFlag( D, true );
    # remember the factors
    D!.factors := [ nr1, nr2 ];

    return D;
  end );

#############################################################################
##
#M  ViewObj   for direct products of nearrings
##

InstallMethod(
ViewObj,
"direct products of nearrings",
true,
[IsNearRing and IsDirectProductNearRing],
10,   # overwrite all methods for ExpMulNRs
  function ( nr )
    Print("DirectProductNearRing( ",nr!.factors[1],", ",nr!.factors[2]," )");
  end );

#############################################################################
##
#M  NearRingIdeals   for direct products of nearrings
##

InstallMethod(
NearRingIdeals,
"direct products",
true,
[IsNearRingWithOne and IsDirectProductNearRing],
10,
  function ( nr )
  local addGroup,
idealsOf1stFactor, idealsOf2ndFactor,
embedding1, embedding2,
ideals, I, J;

    addGroup := GroupReduct(nr);

    idealsOf1stFactor := NearRingIdeals(nr!.factors[1]);
    idealsOf2ndFactor := NearRingIdeals(nr!.factors[2]);

    embedding1 := Embedding( addGroup, 1 );
    embedding2 := Embedding( addGroup, 2 );

    idealsOf1stFactor := List( idealsOf1stFactor,
id -> Image( embedding1, GroupReduct( id ) ) );
    idealsOf2ndFactor := List( idealsOf2ndFactor,
id -> Image( embedding2, GroupReduct( id ) ) );

    ideals := [];
    for I in idealsOf1stFactor do
      for J in idealsOf2ndFactor do
        Add( ideals, ClosureGroup( I, J ) );
      od;
    od;

    return List( ideals, I -> NearRingIdealBySubgroupNC( nr, I ) );
  end );

#############################################################################
##
#M  NearRingLeftIdeals   for direct products of nearrings
##

InstallMethod(
NearRingLeftIdeals,
"direct products",
true,
[IsNearRingWithOne and IsDirectProductNearRing],
10,
  function ( nr )
  local addGroup,
idealsOf1stFactor, idealsOf2ndFactor,
embedding1, embedding2,
ideals, I, J;

    addGroup := GroupReduct(nr);

    idealsOf1stFactor := NearRingLeftIdeals(nr!.factors[1]);
    idealsOf2ndFactor := NearRingLeftIdeals(nr!.factors[2]);

    embedding1 := Embedding( addGroup, 1 );
    embedding2 := Embedding( addGroup, 2 );

    idealsOf1stFactor := List( idealsOf1stFactor,
id -> Image( embedding1, GroupReduct( id ) ) );
    idealsOf2ndFactor := List( idealsOf2ndFactor,
id -> Image( embedding2, GroupReduct( id ) ) );

    ideals := [];
    for I in idealsOf1stFactor do
      for J in idealsOf2ndFactor do
        Add( ideals, ClosureGroup( I, J ) );
      od;
    od;

    return List( ideals, I -> NearRingLeftIdealBySubgroupNC( nr, I ) );
  end );

#############################################################################
##
#M  NearRingRightIdeals   for direct products of nearrings
##

InstallMethod(
NearRingRightIdeals,
"direct products",
true,
[IsNearRingWithOne and IsDirectProductNearRing],
10,
  function ( nr )
  local addGroup,
idealsOf1stFactor, idealsOf2ndFactor,
embedding1, embedding2,
ideals, I, J;

    addGroup := GroupReduct(nr);

    idealsOf1stFactor := NearRingRightIdeals(nr!.factors[1]);
    idealsOf2ndFactor := NearRingRightIdeals(nr!.factors[2]);

    embedding1 := Embedding( addGroup, 1 );
    embedding2 := Embedding( addGroup, 2 );

    idealsOf1stFactor := List( idealsOf1stFactor,
id -> Image( embedding1, GroupReduct( id ) ) );
    idealsOf2ndFactor := List( idealsOf2ndFactor,
id -> Image( embedding2, GroupReduct( id ) ) );

    ideals := [];
    for I in idealsOf1stFactor do
      for J in idealsOf2ndFactor do
        Add( ideals, ClosureGroup( I, J ) );
      od;
    od;

    return List( ideals, I -> NearRingRightIdealBySubgroupNC( nr, I ) );
  end );

#############################################################################
##
#M  FactorNearRing
##

InstallMethod(
FactorNearRing,
"really an ideal?",
IsIdenticalObj,
[IsExplicitMultiplicationNearRing, IsNRI],
100,
  function ( N, I )
  if IsNearRingRightIdeal(I) and
     IsNearRingLeftIdeal(I) then
     TryNextMethod();
  else
     Error( "FactorNearRing( N, I ): I has to be a two-sided ideal in N" );
  fi;
end );

InstallMethod(
FactorNearRing,
"ExpMulNR",
IsIdenticalObj,
[IsExplicitMultiplicationNearRing, IsNRI],
0,
  function ( N, I )
  local NfI, f, mul, addN, addI, F;
    addN := GroupReduct(N); addI := GroupReduct(I);
    f := NaturalHomomorphismByNormalSubgroupNC( addN, addI );
    NfI := Image( f, addN );

    mul := function ( x, y )
      return Image( f,
NRMultiplication(N)(
  PreImagesRepresentative( f, x ),
  PreImagesRepresentative( f, y )
    )
    );
    end;
    F := ExplicitMultiplicationNearRingNC( NfI, mul );
    SetFactorNearRingFlag( F, true );
    F!.mother := N;   # the generators of
    F!.father := I;   # a factor nearring

    return F;
end );

InstallMethod(
FactorNearRing,
"generic",
IsIdenticalObj,
[IsNearRing, IsNRI],
0,
  function ( N, I )
  local NfI, f, mul, addN, addI, fam, F;
    fam := N!.elementsInfo;
    addN := GroupReduct(N); addI := GroupReduct(I);
    f := NaturalHomomorphismByNormalSubgroupNC( addN, addI );
    NfI := Image( f, addN );

    mul := function ( x, y )
      return Image( f,
GroupElementRepOfNearRingElement(
  NearRingElementByGroupRep( fam, PreImagesRepresentative( f, x ) ) *
  NearRingElementByGroupRep( fam, PreImagesRepresentative( f, y ) )
     )
    );
    end;
    F := ExplicitMultiplicationNearRingNC( NfI, mul );
    SetFactorNearRingFlag( F, true );
    F!.mother := N;   # the generators of
    F!.father := I;   # a factor nearring

    return F;
end );

#############################################################################
##
#M  \/    for a nearring and an ideal
##

InstallOtherMethod(
\/,
"nearrings",
true,
[IsNearRing, IsNRI],
0,
  function ( N, I )
    return FactorNearRing( N, I );
  end );

#############################################################################
##
#M  ViewObj   for factor NearRings
##

InstallMethod(
ViewObj,
"factor NearRings",
true,
[IsNearRing and IsFactorNearRing],
10,   # overwrite all methods for ExpMulNRs
  function ( nr )
    Print( "FactorNearRing( ",nr!.mother,", ",nr!.father," )" );
  end );

Quelle nrconstr.gi Sprache: unbekannt

[ Dauer der Verarbeitung: 0.28 Sekunden (vorverarbeitet) ]