Quelle gpnmap.gi
Sprache: unbekannt
|
|
Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
##############################################################################
##
#W gpnmap.gi GAP4package `XMod' Chris Wensley
## Alper Odabas
## This file implements functions for Higher Dimensional Mappings for
## (pre-)catn-groups.
##
#Y Copyright (C) 2001-2024, Chris Wensley et al,
##############################################################################
##
#M MakeHigherDimensionalGroupMorphism( [ <src>, <rng>, <list of maps> )
##
InstallMethod( MakeHigherDimensionalGroupMorphism,
"for two higher dimensional objects and list of 2d-morphisms", true,
[ IsHigherDimensionalGroup, IsHigherDimensionalGroup, IsList ], 0,
function( src, rng, mors )
local mor, dim, n, i;
dim := HigherDimension( src );
if ( dim <> HigherDimension( rng ) ) then
Info( InfoXMod, 1, "Source and Range must have the same dimension" );
return fail;
fi;
n := Length( mors );
if ( HasIsPreCat2Group(src) and IsPreCat2Group(src) ) then
if ( 2^(dim-1) <> n ) then
Info( InfoXMod, 1, "3rd argument should have length ", 2^(dim-1) );
return fail;
fi;
elif ( HasIsPreCrossedSquare(src) and IsPreCrossedSquare(src) ) then
## morphism of (pre)crossed squares
if ( dim <> n-1 ) then
Info( InfoXMod, 1, "third argument should have length ", dim+1 );
return fail;
fi;
fi;
if ForAny( mors, x -> not Is2DimensionalGroupMorphism(x) ) then
Info( InfoXMod, 1,
"Entries in list mors must be pre-cat1-group morphisms" );
return fail;
fi;
mor := rec();
ObjectifyWithAttributes( mor, TypeHigherDimensionalGroupMorphism,
Source, src,
Range, rng,
HigherDimension, dim );
if ( HasIsPreCrossedSquare( src ) and IsPreCrossedSquare( src ) )
or ( HasIsPreCat2Group( src ) and IsPreCat2Group( src ) ) then
SetUp2DimensionalMorphism( mor, mors[1] );
SetLeft2DimensionalMorphism( mor, mors[2] );
SetRight2DimensionalMorphism( mor, mors[3] );
SetDown2DimensionalMorphism( mor, mors[4] );
else
SetListOf2DimensionalMappings( mor, mors );
fi;
return mor;
end );
#############################################################################
##
#M IsPreCatnGroupMorphism . . . . . check the axioms for a pre-catn morphism
##
InstallMethod( IsPreCatnGroupMorphism,
"generic method for pre-catn homomorphisms", true,
[ IsHigherDimensionalGroupMorphism ], 0,
function( mor )
local PC, QC, dim, upmor, dnmor, ok, 2dmor, genPC, genQC, srcPC, rngPC,
srcQC, rngQC, rangeQC, x, rnghoms, srchoms, G1, G2, P1, P2, p, q,
comp, perm2dmor, i, gamma, rho, sigma, pi;
PC := Source( mor );
QC := Range( mor );
dim := HigherDimension( PC );
if not ( dim = HigherDimension( QC ) ) then
Info( InfoXMod, 1, "PC, QC do not have the same dimension" );
return false;
fi;
if ( dim = 3 ) then
if ( HasIsPreCat2Group( PC ) and IsPreCat2Group( PC ) ) then
return IsPreCat2GroupMorphism( mor );
else
return IsPreCrossedSquareMorphism( mor );
fi;
fi;
2dmor := ListOf2DimensionalMappings( mor );
if ForAny( 2dmor, x -> not Is2DimensionalGroupMorphism(x) ) then
Info( InfoXMod, 1, "Is2DimensionalGroupMorphism(x) fails for ", x );
return false;
fi;
genPC := GeneratingCat1Groups( PC );
genQC := GeneratingCat1Groups( QC );
srcPC := List( genPC, Source );
srcQC := List( genQC, Source );
rngPC := List( genPC, Range );
rngQC := List( genQC, Range );
rnghoms := List( 2dmor, RangeHom );
srchoms := List( 2dmor, SourceHom );
for x in [1..Length(genPC)] do
if ( Source( srchoms[x] ) <> srcPC[x] ) then
Info( InfoXMod, 1, x, "a : ", Source(rnghoms[x]),
" <> ", srcPC[x] );
return false;
fi;
if ( Range( srchoms[x] ) <> srcQC[x] ) then
Info( InfoXMod, 1, x, "b : ", Range(srchoms[x]),
" <> ", srcQC[x] );
return false;
fi;
if ( Source( rnghoms[x] ) <> rngPC[x] ) then
Info( InfoXMod, 1, x, "a : ", Source(rnghoms[x]),
" <> ", rngPC[x] );
return false;
fi;
if ( Range( rnghoms[x] ) <> rngQC[x] ) then
Info( InfoXMod, 1, x, "b : ", Source(rnghoms[x]),
" <> ", rngQC[x] );
return false;
fi;
od;
# check equality of source homomorphisms
gamma := SourceHom( 2dmor[1] );
for i in [2..dim-1] do
if not ( gamma = SourceHom( 2dmor[i] ) ) then
Info( InfoXMod, 2, "unequal source homs" );
return false;
fi;
od;
Print( "WARNING: further checks are needed here\n" );
return true;
end );
#############################################################################
##
#M IsCatnGroupMorphism
##
InstallMethod( IsCatnGroupMorphism, "generic method for higher dim mappings",
true, [ IsHigherDimensionalGroupMorphism ], 0,
function( mor )
local ispre;
ispre := IsPreCatnGroupMorphism( mor );
if not ispre then
return false;
else
return ( IsCatnGroup( Source(mor) ) and IsCatnGroup( Range(mor) ) );
fi;
end );
##############################################################################
##
#M \=( <mor>, <phi> ) . . test if two higher dimensional morphisms are equal
##
InstallMethod( \=,
"generic method for two higher dimensional morphisms", IsIdenticalObj,
[ IsHigherDimensionalMapping, IsHigherDimensionalMapping ], 0,
function ( mor, phi )
local n1, n2, mors1, mors2;
n1 := HigherDimension( mor );
n2 := HigherDimension( phi );
if ( n1 <> n2 ) then
return false;
fi;
if not ( ( Source(mor) = Source(phi) ) and
( Range(mor) = Range(phi) ) ) then
return false;
fi;
if ( n1 = 3 ) then
return ( ( Up2DimensionalMorphism( mor )
= Up2DimensionalMorphism( phi ) )
and ( Left2DimensionalMorphism( mor )
= Left2DimensionalMorphism( phi ) )
and ( Right2DimensionalMorphism( mor )
= Right2DimensionalMorphism( phi ) )
and ( Down2DimensionalMorphism( mor )
= Down2DimensionalMorphism( phi ) ) );
else
mors1 := ListOf2DimensionalMappings( mor );
mors2 := ListOf2DimensionalMappings( phi );
return ( mors1 = mors2 );
fi;
end );
#############################################################################
##
#F MappingGeneratorsImages( <map> ) . . . . . for a higher dimensional mapping
##
InstallOtherMethod( MappingGeneratorsImages, "method for a higher dim mapping",
true, [ IsHigherDimensionalMapping ], 0,
function( map )
local mors, imors;
mors := ListOf2DimensionalMappings( map );
imors := List( mors, MappingGeneratorsImages );
return imors;
end );
#############################################################################
##
#M Name . . . . . . . . . . . . . . . . . . or a higher dimensional mapping
##
InstallMethod( Name, "method for a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
function( mor )
local nsrc, nrng, name;
if HasName( Source( mor ) ) then
nsrc := Name( Source( mor ) );
else
nsrc := "[..]";
fi;
if HasName( Range( mor ) ) then
nrng := Name( Range( mor ) );
else
nrng := "[..]";
fi;
name := Concatenation( "[", nsrc, " => ", nrng, "]" );
SetName( mor, name );
return name;
end );
#############################################################################
##
#F Display( <mor> ) . . . . . print details of a higher dimensional mapping
##
InstallMethod( Display, "display a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
function( mor )
local P, Q, n, mors, i;
P := Source( mor );
Q := Range( mor );
n := HigherDimension( mor );
if (( n = 3 ) and HasIsPreCrossedSquareMorphism( mor ) ) then
Display3DimensionalMorphism( mor );
else
mors := ListOf2DimensionalMappings( mor );
if ( HasIsCatnGroupMorphism( mor ) and IsCatnGroupMorphism( mor ) ) then
Print( "Morphism of cat",n-1,"-groups :- \n" );
else
Print( "Morphism of pre-cat",n-1,"-groups :- \n" );
fi;
if HasName( P ) then
Print( ": Source = ", Name( P ), "\n" );
else
Print( ": Source has ", P, "\n" );
fi;
if HasName( Q ) then
Print( ": Range = ", Name( Q ), "\n" );
else
Print( ": Range has ", Q, "\n" );
fi;
if HasOrder( mor ) then
Print( ": order = ", Order( mor ), "\n" );
fi;
Print( ": MappingGeneratorsImages for the source homomorphisms:\n" );
for i in [1..Length(mors)] do
Print( i, " : ", MappingGeneratorsImages(SourceHom(mors[i])),"\n");
od;
fi;
end );
##############################################################################
##
#M IdentityMapping( <obj> )
##
InstallOtherMethod( IdentityMapping, "for higher dimensional object", true,
[ IsHigherDimensionalGroup ], 0,
function( obj )
local dim, idmaps;
## this works for catn-groups but should be extended to n-cubes
dim := HigherDimension( obj );
if ( dim < 4 ) then
Error( "should be using method for low dimensions" );
else
idmaps := List( GeneratingCat1Groups( obj ), IdentityMapping );
return PreCatnGroupMorphismByMorphisms( obj, obj, idmaps );
fi;
end );
##############################################################################
##
#F CatnGroupMorphism( [list] ) . . catn morphism
##
InstallGlobalFunction( CatnGroupMorphism, function( arg )
local nargs;
nargs := Length( arg );
if ( ( nargs = 3 ) and IsCatnGroup( arg[1])
and IsCatnGroup( arg[2] ) and IsList( arg[3] ) ) then
return CatnGroupMorphismByMorphisms( arg[1], arg[2], arg[3] );
fi;
# alternatives not allowed
Info( InfoXMod, 2, "usage: CatnGroupMorphism( src, rng, list of maps );" );
return fail;
end );
###############################################################################
##
#M PreCatnGroupMorphismByMorphisms( <src>, <rng>, <list of maps> )
##
InstallMethod( PreCatnGroupMorphismByMorphisms,
"method for a higher dimensional mapping", true,
[ IsPreCatnGroup, IsPreCatnGroup, IsList ], 0,
function( src, rng, L )
local dim, filter, fam, mor, ok, nsrc, nrng, name, n;
dim := HigherDimension( src )-1;
if not ( dim = HigherDimension(rng)-1 ) then
Error( "src and rng have different dimensions" );
fi;
if ( Length(L) <> dim ) then
Info( InfoXMod, 1, "L should contain ", dim, " morphisms" );
return fail;
fi;
mor := MakeHigherDimensionalGroupMorphism( src, rng, L );
if not IsPreCatnGroupMorphism( mor ) then
Info( InfoXMod, 2, "not a morphism of pre-catn-groups" );
return fail;
fi;
return mor;
end );
##############################################################################
##
#M CatnGroupMorphismByMorphisms . . . . . . for a higher dimensional mapping
##
InstallMethod( CatnGroupMorphismByMorphisms, "method for a nd-mapping", true,
[ IsCatnGroup, IsCatnGroup, IsList], 0,
function( src, rng, L )
local mor, ok;
mor := PreCatnGroupMorphismByMorphisms( src, rng, L );
ok := IsCatnGroupMorphism( mor );
if not ok then
return fail;
fi;
return mor;
end );
##############################################################################
##
#M CompositionMorphism . . . . . . . . . . .for a higher dimensional mapping
##
InstallOtherMethod( CompositionMorphism, "generic method for nd-mappings",
IsIdenticalObj, [ IsHigherDimensionalMapping, IsHigherDimensionalMapping ],
0,
function( m2, m1 )
local up, lt, rt, dn, comp, ok, list, comps, n1, n2, mors1, mors2, i;
n1 := HigherDimension( m1 );
n2 := HigherDimension( m2 );
mors1 := ListOf2DimensionalMappings( m1 );
mors2 := ListOf2DimensionalMappings( m2 );
if not ( Range( m1 ) = Source( m2 ) ) then
Info( InfoXMod, 2, "Range(m1) <> Source(m2)" );
return fail;
fi;
if not ( n1 = n2 ) then
Info( InfoXMod, 2, "Dimension - (m1) <> Dimension - (m2)" );
return fail;
fi;
comps := List( [1..Length(mors1)],
i -> CompositionMapping( mors2[i], mors1[i] ) );
list := [ Source( m1 ), Range( m2 ) ];
Append( list, comps );
comp := MakeHigherDimensionalMapping( list );
if ( IsPreCatnGroupMorphism( m1 ) and IsPreCatnGroupMorphism( m2 ) ) then
SetIsPreCatnGroupMorphism( comp, true );
fi;
if ( IsCatnGroupMorphism( m1 ) and IsCatnGroupMorphism( m2 ) ) then
SetIsCatnGroupMorphism( comp, true );
fi;
return comp;
end );
##############################################################################
##
#M Order . . . . . . . . . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( Order, "method for a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
function( mor )
local dim, ok, 2d_maps;
dim := HigherDimension( mor );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(mor), Left2DimensionalMorphism(mor),
Right2DimensionalMorphism(mor), Down2DimensionalMorphism(mor) ];
else
2d_maps := ListOf2DimensionalMappings(mor);
fi;
if not ( IsAutomorphismHigherDimensionalDomain(mor) ) then
Info( InfoXMod, 2, "Parameter is not an automorphism" );
return fail;
fi;
return Lcm( List(2d_maps, Order ) );
end );
##############################################################################
##
#M IsInjective( map ) . . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( IsInjective, "method for a higher dimensional mapping",
true, [ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsInjective );
if not ok then
return false;
fi;
return true;
end );
##############################################################################
##
#M IsSurjective( map ) . . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( IsSurjective, "method for a higher dimensional mapping",
true, [ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsSurjective );
if not ok then
return false;
fi;
return true;
end );
##############################################################################
##
#M IsSingleValued( map ) . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( IsSingleValued, "method for a higher dimensional mapping",
true, [ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsSingleValued );
if not ok then
return false;
fi;
return true;
end );
##############################################################################
##
#M IsTotal( map ) . . . . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( IsTotal, "method for a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsTotal );
if not ok then
return false;
fi;
return true;
end );
##############################################################################
##
#M IsBijective( map ) . . . . . . . . . . . for a higher dimensional mapping
##
InstallOtherMethod( IsBijective, "method for a higher dimensional mapping",
true, [ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsBijective );
if not ok then
return false;
fi;
return true;
end );
##############################################################################
##
#M IsEndomorphismHigherDimensionalDomain( map ) . . for a higher dim mapping
#M IsAutomorphismHigherDimensionalDomain( map ) . . for a higher dim mapping
##
InstallMethod( IsEndomorphismHigherDimensionalDomain,
"method for a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
function( map )
local dim, ok, 2d_maps;
dim := HigherDimension( map );
if ( dim = 3 ) then
2d_maps :=
[ Up2DimensionalMorphism(map), Left2DimensionalMorphism(map),
Right2DimensionalMorphism(map), Down2DimensionalMorphism(map) ];
else
2d_maps := ListOf2DimensionalMappings(map);
fi;
ok := ForAll( 2d_maps, IsEndomorphism2DimensionalDomain );
if not ok then
return false;
fi;
return true;
end );
InstallMethod( IsAutomorphismHigherDimensionalDomain,
"method for a higher dimensional mapping", true,
[ IsHigherDimensionalMapping ], 0,
map -> IsEndomorphismHigherDimensionalDomain(map) and IsBijective(map) );
[ Dauer der Verarbeitung: 0.115 Sekunden
]
|
2026-03-28
|
