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#############################################################################
##
#M ClosureNearRingIdeal
InstallMethod(
ClosureNearRingIdeal,
"nrid + nrid",
IsIdenticalObj,
[IsNRI and IsNearRingIdeal,IsNRI and IsNearRingIdeal],
2,
function ( i, j )
local sum;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
sum := ClosureGroup( GroupReduct(i), GroupReduct(j) );
sum := NearRingIdealBySubgroupNC( Parent(i), sum );
if HasGeneratorsOfNearRingIdeal(i) and HasGeneratorsOfNearRingIdeal(j) then
SetGeneratorsOfNearRingIdeal( sum,
Concatenation( GeneratorsOfNearRingIdeal(i),
GeneratorsOfNearRingIdeal(j) ) );
fi;
return sum;
end );
#############################################################################
##
#M ClosureNearRingLeftIdeal
InstallMethod(
ClosureNearRingLeftIdeal,
"nrLeftId + nrLeftId",
IsIdenticalObj,
[IsNRI and IsNearRingLeftIdeal,
IsNRI and IsNearRingLeftIdeal],
1,
function ( i, j )
local sum;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
sum := ClosureGroup( GroupReduct(i), GroupReduct(j) );
sum := NearRingLeftIdealBySubgroupNC( Parent(i), sum );
if HasGeneratorsOfNearRingLeftIdeal(i) and
HasGeneratorsOfNearRingLeftIdeal(j) then
SetGeneratorsOfNearRingLeftIdeal( sum,
Concatenation( GeneratorsOfNearRingLeftIdeal(i),
GeneratorsOfNearRingLeftIdeal(j) ) );
fi;
return sum;
end );
#############################################################################
##
#M ClosureNearRingRightIdeal
InstallMethod(
ClosureNearRingRightIdeal,
"nrRightId + nrRightId",
IsIdenticalObj,
[IsNRI and IsNearRingRightIdeal,
IsNRI and IsNearRingRightIdeal],
0,
function ( i, j )
local sum;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
sum := ClosureGroup( GroupReduct(i), GroupReduct(j) );
sum := NearRingRightIdealBySubgroupNC( Parent(i), sum );
if HasGeneratorsOfNearRingRightIdeal(i) and
HasGeneratorsOfNearRingRightIdeal(j) then
SetGeneratorsOfNearRingRightIdeal( sum,
Concatenation( GeneratorsOfNearRingRightIdeal(i),
GeneratorsOfNearRingRightIdeal(j) ) );
fi;
return sum;
end );
#############################################################################
##
#M Intersection2
InstallMethod(
Intersection2,
"two nearring ideals",
IsIdenticalObj,
[IsNRI and IsNearRingIdeal,
IsNRI and IsNearRingIdeal],
2,
function ( i, j )
local intersect;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
intersect := Intersection( GroupReduct(i), GroupReduct(j) );
return NearRingIdealBySubgroupNC( Parent(i), intersect );
end );
InstallMethod(
Intersection2,
"two nearring LeftIdeals",
IsIdenticalObj,
[IsNRI and IsNearRingLeftIdeal,
IsNRI and IsNearRingLeftIdeal],
1,
function ( i, j )
local intersect;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
intersect := Intersection( GroupReduct(i), GroupReduct(j) );
return NearRingLeftIdealBySubgroupNC( Parent(i), intersect );
end );
InstallMethod(
Intersection2,
"two nearring RightIdeals",
IsIdenticalObj,
[IsNRI and IsNearRingRightIdeal,
IsNRI and IsNearRingRightIdeal],
0,
function ( i, j )
local intersect;
if Parent(i)<>Parent(j) then
Error("the two ideals must have equal parents");
fi;
intersect := Intersection( GroupReduct(i), GroupReduct(j) );
return NearRingRightIdealBySubgroupNC( Parent(i), intersect );
end );
#############################################################################
##
#M IsSimpleNearRing
##
InstallTrueMethod( IsSimpleNearRing, IsFullTransformationNearRing );
InstallTrueMethod( IsSimpleNearRing, IsNearField );
InstallMethod(
IsSimpleNearRing,
"small nearrings", # test principal ideals
true,
[IsNearRing],
1,
function ( N )
if Size( N ) > 1000 then
TryNextMethod();
else
return ForAll( List( RationalClasses( GroupReduct(N) ), Representative ), x ->
Size( NearRingIdealClosureOfSubgroup( N, NormalClosure( GroupReduct(N), Subgroup( GroupReduct(N), [x] ) ) ) ) in [1,Size(N)] );
fi;
end );
InstallMethod(
IsSimpleNearRing,
"known ideals", # count ideals
true,
[IsNearRing and HasNearRingIdeals],
1,
function ( N )
return Length(NearRingIdeals(N)) = 2;
end );
InstallMethod(
IsSimpleNearRing,
"default", # count ideals
true,
[IsNearRing],
0,
function ( N )
return Length(NearRingIdeals(N)) = 2;
end );
#############################################################################
##
#M NearRingCommutator
InstallMethod(
NearRingCommutator,
"first",
IsIdenticalObj,
[IsNRI and IsNearRingIdeal, IsNRI and IsNearRingIdeal],
0,
function ( X, Y )
local N,d,d2,comm, m;
N := Parent(X);
if N<>Parent(Y) or not IsNearRing(N) then return fail; fi;
if AdditiveGenerators(X)=[] or AdditiveGenerators(Y)=[] then
return NearRingIdealByGenerators(N,[]);
elif IsNearRingWithOne(N) then
if N=X then
return Y;
elif N=Y then
return X;
fi;
fi;
d:= Union(
Union(List(N,
a->Union(List(AdditiveGenerators(X),
x->List(AdditiveGenerators(Y),
y->(a+y)*x-a*x))))),
Union(List(N,
a->Union(List(AdditiveGenerators(X),
x->List(AdditiveGenerators(Y),
y->(a+x)*y-a*y))))),
Union(List(X,
x->List(Y,
y-> x+y-x-y)))
);
d2:=Union(List(AdditiveGenerators(N),
a->Union(List(X,
x->Union(List(Y,
y->List(N,
b->(b+x+y)*a - (b+x)*a + b*a - (b+y)*a)))))));
return NearRingIdealByGenerators(N, Union(d,d2));
end );
#############################################################################
##
#M NearRingCommutatorsTable
InstallMethod(
NearRingCommutatorsTable,
"default",
true,
[IsNearRing],
0,
function(N)
local ids,i;
ids:=NearRingIdeals(N);
return List(ids,
ix->List(ids,
iy->Position(ids,NearRingCommutator(ix,iy))));
end );
#############################################################################
##
#M PrintNearRingCommutatorsTable
## As above, but for better printing (to be improved)
InstallMethod(
PrintNearRingCommutatorsTable,
"default",
true,
[IsNearRing],
0,
function(N)
local line;
for line in NearRingCommutatorsTable(N) do
Print(line,"\n");
od;
end );
[ Dauer der Verarbeitung: 0.21 Sekunden
(vorverarbeitet)
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