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## 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 );
[ Dauer der Verarbeitung: 0.37 Sekunden
(vorverarbeitet)
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