Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
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##
#W double.tst Groupoids Package Chris Wensley
##
gap> START_TEST( "groupoids package: double.tst" );
gap> gpd_infolevel_saved := InfoLevel( InfoGroupoids );;
gap> SetInfoLevel( InfoGroupoids, 0 );;
## make double.tst independent of gpd.tst and gpdhom.tst
gap> a4 := Group( (1,2,3), (2,3,4) );;
gap> d8 := Group( (5,6,7,8), (5,7) );;
gap> SetName( a4, "a4" ); SetName( d8, "d8" );
gap> Ga4 := SinglePieceGroupoid( a4, [-15 .. -11] );;
gap> Gd8 := Groupoid( d8, [-9,-8,-7] );;
gap> c6 := Group( (11,12,13)(14,15) );;
gap> SetName( c6, "c6" );
gap> Gc6 := MagmaWithSingleObject( c6, -10 );;
gap> SetName( Ga4, "Ga4" ); SetName( Gd8, "Gd8" ); SetName( Gc6, "Gc6" );
gap> q8 := QuaternionGroup( 8 );;
gap> genq8 := GeneratorsOfGroup( q8 );;
gap> x := genq8[1];; y := genq8[2];;
gap> Gq8 := Groupoid( q8, [ -19, -18, -17 ] );;
gap> SetName( q8, "q8" ); SetName( Gq8, "Gq8" );
gap> U3 := UnionOfPieces( [ Ga4, Gc6, Gd8 ] );;
## SubSection 8.1.1
gap> DGd8 := SinglePieceBasicDoubleGroupoid( Gd8 );;
gap> DGd8!.groupoid;
Gd8
gap> DGd8!.objects;
[ -9, -8, -7 ]
gap> SetName( DGd8, "DGd8" );
gap> [ IsDoubleGroupoid( DGd8 ), IsBasicDoubleGroupoid( DGd8 ) ];
[ true, true ]
## SubSection 8.1.2
gap> [ Size(DGd8), (3*8)^4 ];
[ 331776, 331776 ]
gap> a1 := Arrow( Gd8, (5,7), -7, -8 );;
gap> b1 := Arrow( Gd8, (6,8), -7, -7 );;
gap> c1 := Arrow( Gd8, (5,6)(7,8), -8, -9 );;
gap> d1 := Arrow( Gd8, (5,6,7,8), -7, -9 );;
gap> bdy1 := d1^-1 * b1^-1 * a1 * c1;
[(6,8) : -9 -> -9]
gap> sq1 := SquareOfArrows( DGd8, a1, b1, c1, d1 );
[-7] ------- (5,7) ------> [-8]
| |
(6,8) (6,8) (5,6)(7,8)
V V
[-7] ----- (5,6,7,8) ----> [-9]
gap> sq1 in DGd8;
true
gap> UpArrow( sq1 );
[(5,7) : -7 -> -8]
gap> LeftArrow( sq1 );
[(6,8) : -7 -> -7]
gap> RightArrow( sq1 );
[(5,6)(7,8) : -8 -> -9]
gap> DownArrow( sq1 );
[(5,6,7,8) : -7 -> -9]
gap> BoundaryOfSquare( sq1 );
[(6,8) : -9 -> -9]
gap> DoubleGroupoidOfSquare( sq1 );
DGd8
gap> IsDoubleGroupoidElement( sq1 );
true
## SubSection 8.1.3
gap> a2 := Arrow( Gd8, (6,8), -8, -9 );;
gap> c2 := Arrow( Gd8, (5,7)(6,8), -9, -8);;
gap> d2 := Arrow( Gd8, (5,6,7,8), -9, -8 );;
gap> sq2 := SquareOfArrows( DGd8, a2, c1, c2, d2 );
[-8] -------- (6,8) -------> [-9]
| |
(5,6)(7,8) () (5,7)(6,8)
V V
[-9] ------ (5,6,7,8) -----> [-8]
gap> bdy2 := BoundaryOfSquare( sq2 );
[() : -8 -> -8]
gap> [ IsCommutingSquare(sq1), IsCommutingSquare(sq2) ];
[ false, true ]
## SubSection 8.1.4
gap> tsq1 := TransposedSquare( sq1 );
[-7] ------- (6,8) ------> [-7]
| |
(5,7) (6,8) (5,6,7,8)
V V
[-8] ---- (5,6)(7,8) ---> [-9]
gap> IsClosedUnderTransposition( sq1 );
false
## SubSection 8.1.5
gap> LeftArrow( sq2 ) = RightArrow( sq1 );
true
gap> sq12 := HorizontalProduct( sq1, sq2 );
[-7] ----- (5,7)(6,8) ----> [-9]
| |
(6,8) (5,7) (5,7)(6,8)
V V
[-7] ----- (5,7)(6,8) ----> [-8]
gap> bdy12 := BoundaryOfSquare( sq12 );
[(5,7) : -8 -> -8]
gap> (bdy1^d2) * bdy2 = bdy12;
true
## SubSection 8.1.6
gap> b3 := Arrow( Gd8, (5,7), -7, -9 );;
gap> c3 := Arrow( Gd8, (6,8), -9, -8);;
gap> d3 := Arrow( Gd8, (5,8)(6,7), -9, -8 );;
gap> sq3 := SquareOfArrows( DGd8, d1, b3, c3, d3 );
[-7] ---- (5,6,7,8) ---> [-9]
| |
(5,7) (6,8) (6,8)
V V
[-9] ---- (5,8)(6,7) ---> [-8]
gap> bdy3 := BoundaryOfSquare( sq3 );
[(6,8) : -8 -> -8]
gap> UpArrow( sq3 ) = DownArrow( sq1 );
true
gap> sq13 := VerticalProduct( sq1, sq3 );
[-7] -------- (5,7) -------> [-8]
| |
(5,7)(6,8) () (5,8,7,6)
V V
[-9] ----- (5,8)(6,7) ----> [-8]
gap> c4 := Arrow( Gd8, (5,6,7,8), -8, -7);;
gap> d4 := Arrow( Gd8, (5,6)(7,8), -8, -7 );;
gap> sq4 := SquareOfArrows( DGd8, d2, c3, c4, d4 );
[-9] ------- (5,6,7,8) ------> [-8]
| |
(6,8) (5,6,7,8) (5,6,7,8)
V V
[-8] ------ (5,6)(7,8) -----> [-7]
gap> UpArrow( sq4 ) = DownArrow( sq2 );
true
gap> LeftArrow( sq4 ) = RightArrow( sq3 );
true
gap> sq34 := HorizontalProduct( sq3, sq4 );
[-7] ------- (5,7)(6,8) ------> [-8]
| |
(5,7) (5,8)(6,7) (5,6,7,8)
V V
[-9] ------- (5,7)(6,8) ------> [-7]
gap> sq1234 := VerticalProduct( sq12, sq34 );
[-7] --------- (5,7)(6,8) --------> [-9]
| |
(5,7)(6,8) (5,6,7,8) (5,8,7,6)
V V
[-9] --------- (5,7)(6,8) --------> [-7]
gap> sq24 := VerticalProduct( sq2, sq4 );
[-8] ----------- (6,8) ----------> [-9]
| |
(5,8,7,6) (5,6,7,8) (5,8,7,6)
V V
[-8] -------- (5,6)(7,8) -------> [-7]
gap> sq1324 := HorizontalProduct( sq13, sq24 );;
gap> sq1324 = sq1234;
true
## SubSection 8.1.7
gap> hid := HorizontalIdentities( sq24 );;
gap> hid[1]; Print("\n"); hid[2];
[-8] --------- () --------> [-8]
| |
(5,8,7,6) () (5,8,7,6)
V V
[-8] --------- () --------> [-8]
[-9] --------- () --------> [-9]
| |
(5,8,7,6) () (5,8,7,6)
V V
[-7] --------- () --------> [-7]
gap> HorizontalProduct( hid[1], sq24 ) = sq24;
true
gap> HorizontalProduct( sq24, hid[2] ) = sq24;
true
gap> vid := VerticalIdentities( sq24 );;
gap> vid[1]; Print("\n"); vid[2];
[-8] ---- (6,8) ---> [-9]
| |
() () ()
V V
[-8] ---- (6,8) ---> [-9]
[-8] ---- (5,6)(7,8) ---> [-7]
| |
() () ()
V V
[-8] ---- (5,6)(7,8) ---> [-7]
gap> VerticalProduct( vid[1], sq24 ) = sq24;
true
gap> VerticalProduct( sq24, vid[2] ) = sq24;
true
gap> hinv := HorizontalInverse( sq24 );
[-9] ----------- (6,8) ----------> [-8]
| |
(5,8,7,6) (5,6,7,8) (5,8,7,6)
V V
[-7] -------- (5,6)(7,8) -------> [-8]
gap> HorizontalProduct( hinv, sq24 ) = hid[2];
true
gap> HorizontalProduct( sq24, hinv ) = hid[1];
true
gap> vinv := VerticalInverse( sq24 );
[-8] -------- (5,6)(7,8) -------> [-7]
| |
(5,6,7,8) (5,8,7,6) (5,6,7,8)
V V
[-8] ----------- (6,8) ----------> [-9]
gap> VerticalProduct( vinv, sq24 ) = vid[2];
true
gap> VerticalProduct( sq24, vinv ) = vid[1];
true
## Section 8.2
## SubSection 8.2.1
gap> DGc6 := SinglePieceBasicDoubleGroupoid( Gc6 );;
gap> DGa4 := SinglePieceBasicDoubleGroupoid( Ga4 );;
gap> DGc6s4 := DoubleGroupoid( [ DGc6, DGa4 ] );
double groupoid having 2 pieces :-
1: single piece double groupoid with:
groupoid = Ga4
group = a4
objects = [ -15 .. -11 ]
2: single piece double groupoid with:
groupoid = Gc6
group = c6
objects = [ -10 ]
gap> DGa4c6 := DoubleGroupoid( [ Ga4, Gc6 ] );;
gap> Pieces( DGa4c6 );
[ single piece double groupoid with:
groupoid = Ga4
group = a4
objects = [ -15 .. -11 ], single piece double groupoid with:
groupoid = Gc6
group = c6
objects = [ -10 ] ]
## Section 8.3
## SubSection 8.3.1
gap> DGtriv := DoubleGroupoidWithTrivialGroup( [-19..-17] );
single piece double groupoid with:
groupoid = single piece groupoid: < Group( [ () ] ), [ -19 .. -17 ] >
group = Group( [ () ] )
objects = [ -19 .. -17 ]
gap> Size(DGtriv);
81
## SubSection 8.3.2
gap> DGc4 := DoubleGroupoidWithSingleObject( Group((1,2,3,4)), 0 );
single piece double groupoid with:
groupoid = single piece groupoid: < Group( [ (1,2,3,4) ] ), [ 0 ] >
group = Group( [ (1,2,3,4) ] )
objects = [ 0 ]
gap> Size( DGc4 );
256
## Section 8.4
gap> Gd8c6 := DirectProduct( Gd8, Gc6 );
single piece groupoid: < Group( [ (1,2,3,4), (1,3), (5,6,7)(8,9) ] ),
[ [ -9, -10 ], [ -8, -10 ], [ -7, -10 ] ] >
gap> SetName( Gd8c6, "Gd8c6" );
gap> DGd8c6 := SinglePieceBasicDoubleGroupoid( Gd8c6 );
single piece double groupoid with:
groupoid = Gd8c6
group = Group( [ (1,2,3,4), (1,3), (5,6,7)(8,9) ] )
objects = [ [ -9, -10 ], [ -8, -10 ], [ -7, -10 ] ]
gap> emb1 := Embedding( Gd8c6, 1 );;
gap> emb2 := Embedding( Gd8c6, 2 );;
gap> a5 := Arrow( Gd8, (5,7), -9, -7 );;
gap> a6 := ImageElm( emb1, a5 );
[(1,3) : [ -9, -10 ] -> [ -7, -10 ]]
gap> d5 := Arrow( Gd8, (6,8), -9, -8 );;
gap> d6 := ImageElm( emb1, d5 );
[(2,4) : [ -9, -10 ] -> [ -8, -10 ]]
gap> b5 := Arrow( Gc6, (11,12,13), -10, -10 );;
gap> b6 := ImageElm( emb2, b5 );
[(5,6,7) : [ -9, -10 ] -> [ -9, -10 ]]
gap> c6 := Arrow( Gd8c6, (8,9), [-7,-10], [-8,-10] );;
gap> sq := SquareOfArrows( DGd8c6, a6, b6, c6, d6 );
[[ -9, -10 ]] ----- (1,3) ----> [[ -7, -10 ]]
| |
(5,6,7) (1,3)(2,4)(5,7,6)(8,9) (8,9)
V V
[[ -9, -10 ]] ----- (2,4) ----> [[ -8, -10 ]]
## SubSection 8.5.1
gap> ad8 := GroupHomomorphismByImages( d8, d8,
> [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (6,8) ] );;
gap> md8 := GroupoidHomomorphism( Gd8, Gd8, ad8,
> [-7,-9,-8], [(),(5,7),(6,8)] );;
gap> endDGd8 := DoubleGroupoidHomomorphism( DGd8, DGd8, md8 );;
gap> Display( endDGd8 );
double groupoid homomorphism: [ DGd8 ] -> [ DGd8 ]
with underlying groupoid homomorphism:
homomorphism to single piece groupoid: Gd8 -> Gd8
root group homomorphism:
(5,6,7,8) -> (5,8,7,6)
(5,7) -> (6,8)
object map: [ -9, -8, -7 ] -> [ -7, -9, -8 ]
ray images: [ (), (5,7), (6,8) ]
gap> IsDoubleGroupoidHomomorphism( endDGd8 );
true
gap> sq1;
[-7] ------- (5,7) ------> [-8]
| |
(6,8) (6,8) (5,6)(7,8)
V V
[-7] ----- (5,6,7,8) ----> [-9]
gap> ImageElm( endDGd8, sq1 );
[-8] ------- (5,7) ------> [-9]
| |
(5,7) (5,7) (5,8,7,6)
V V
[-8] ---- (5,8)(6,7) ---> [-7]
gap> DGU3 := DoubleGroupoid( U3 );
double groupoid having 3 pieces :-
1: single piece double groupoid with:
groupoid = Ga4
group = a4
objects = [ -15 .. -11 ]
2: single piece double groupoid with:
groupoid = Gc6
group = c6
objects = [ -10 ]
3: single piece double groupoid with:
groupoid = Gd8
group = d8
objects = [ -9, -8, -7 ]
gap> SetInfoLevel( InfoGroupoids, gpd_infolevel_saved );;
gap> STOP_TEST( "double.tst", 10000 );
