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#############################################################################
##
#W gpgpd.gi GAP4 package `XMod' Chris Wensley
##
#Y Copyright (C) 2001-2024, Chris Wensley et al,
#############################################################################
##
#M GroupGroupoidElement( <cat1> <obj> <obj> ) . . . for a cat1-group element
##
InstallMethod( GroupGroupoidElement, "for a precat1-group and an element",
true, [ IsPreCat1Group, IsObject, IsObject ], 0,
function( C, r, e )
local te, he, tid, hid, elt;
te := ImageElm( TailMap( C ), e );
he := ImageElm( HeadMap( C ), e );
tid := ImageElm( RangeEmbedding( C ), te );
hid := ImageElm( RangeEmbedding( C ), he );
elt := rec( precat1 := C, root := r, element := e,
tail := te, tailid := tid, head := he, headid := hid );
ObjectifyWithAttributes( elt, IsGroupGroupoidElementType,
IsGroupGroupoidElement, true );
return elt;
end );
#############################################################################
##
#M PrintObj( <e> ) . . . . . . . . . . . . . . for a group-groupoid element
##
InstallMethod( PrintObj, "for a group-groupoid element", true,
[ IsGroupGroupoidElement ], 0,
function( e )
Print( e!.tail, ">-", e!.element, "->", e!.head );
end );
#############################################################################
##
#M \*( <e1> <e2> ) . . . . . . . . . . . . . for two group-groupoid elements
##
InstallMethod( \*, "for two group-groupoid elements", true,
[ IsGroupGroupoidElement, IsGroupGroupoidElement ], 0,
function( e1, e2 )
local id, e;
if ( e1!.precat1 <> e2!.precat1 ) or ( e1!.head <> e2!.tail ) then
return fail;
fi;
id := e1!.headid;
e := e1!.element * id^(-1) * e2!.element;
return GroupGroupoidElement( e1!.precat1, e1!.root, e );
end );
#############################################################################
##
#M \=( <e1> <e2> ) . . . . . . . . . . . . . for two group-groupoid elements
##
InstallMethod( \=, "for two group-groupoid elements", true,
[ IsGroupGroupoidElement, IsGroupGroupoidElement ], 0,
function( e1, e2 )
return ( e1!.precat1 = e2!.precat1 ) and ( e1!.element = e2!.element );
end );
#############################################################################
##
#M \<( <e1> <e2> ) . . . . . . . . . . . . . for two group-groupoid elements
##
InstallMethod( \<, "for two group-groupoid elements", true,
[ IsGroupGroupoidElement, IsGroupGroupoidElement ], 0,
function( e1, e2 )
return ( e1!.precat1 = e2!.precat1 ) and ( e1!.element < e2!.element );
end );
#############################################################################
##
#M OneOp( <e> ) . . . . . . . . . . . . . . . . for a group-groupoid element
##
InstallMethod( OneOp, "for a group-groupoid element", true,
[ IsGroupGroupoidElement ], 0,
function( e )
local C, r, g;
C := e!.precat1;
r := e!.root;
g := ImageElm( RangeEmbedding( C ), r );
return GroupGroupoidElement( C, e!.root, g );
end );
#############################################################################
##
#M InverseOp( <e> ) . . . . . . . . . . . . . . for a group-groupoid element
##
InstallMethod( InverseOp, "for a group-groupoid element", true,
[ IsGroupGroupoidElement ], 0,
function( e )
local ig;
ig := e!.headid * e!.element^(-1) * e!.tailid;
return GroupGroupoidElement( e!.precat1, e!.root, ig );
end );
#############################################################################
##
#M GroupGroupoid( <cat1> ) . . . . . . . . . . . . . . for a pre-cat1-group
##
InstallMethod( GroupGroupoid, "for a precat1-group", true,
[ IsPreCat1Group ], 0,
function( C )
local R, G, t, h, e, kert, kerh, kerth, genkerth, nloops, bdy, imbdy,
cosbdy, repbdy, np, eR, pieces, obs, root, eroot, loopgp, cosgp,
repsgp, L1, rays1, size, i, rays, gpd;
R := Range( C );
G := Source( C );
t := TailMap( C );
h := HeadMap( C );
e := RangeEmbedding( C );
kert := Kernel( t );
kerh := Kernel( h );
kerth := Intersection( kert, kerh );
genkerth := GeneratorsOfGroup( kerth );
if ( genkerth = [ ] ) then
genkerth := [ One( G ) ];
fi;
Info( InfoXMod, 1, "genkerth: ", genkerth );
nloops := Size( kerth );
bdy := RestrictedMapping( h, kert );
imbdy := Image( bdy );
cosbdy := RightCosets( R, imbdy );
repbdy := List( cosbdy, Representative );
np := Length( repbdy );
Info( InfoXMod, 1, "repbdy = ", repbdy );
eR := Image( e, R );
pieces := ListWithIdenticalEntries( np, 0 );
Info( InfoXMod, 1, "-------- constructing the first piece --------" );
root := One( R );
eroot := One( G );
loopgp := List( genkerth, g -> GroupGroupoidElement( C, root, g ) );
Info( InfoXMod, 1, "loopgp generators: ", loopgp );
loopgp := GroupWithGenerators( loopgp );
cosgp := RightCosets( kert, kerth );
repsgp := List( cosgp, Representative );
L1 := List( repsgp, g -> g*eroot );
obs := List( repsgp, g -> ImageElm( h, g ) );
rays1 := List( L1, g -> GroupGroupoidElement( C, root, g ) );
Info( InfoXMod, 1, "root = ", root, ", obs = ", obs );
pieces[1] := SinglePieceGroupoidWithRays( loopgp, obs, rays1 );
size := Size( pieces[1] );
## now for the remaining pieces
for i in [2..np] do
Info( InfoXMod, 1, "-------- constructing piece ", i, " --------" );
obs := Elements( cosbdy[i] );
Info( InfoXMod, 1, "obs = ", obs );
root := repbdy[i];
eroot := ImageElm( e, root );
Info( InfoXMod, 1, "root = ", root, ", eroot = ", eroot );
loopgp := List( genkerth, g -> g*eroot );
Info( InfoXMod, 1, "(1) loopgp generators: ", loopgp );
loopgp := List( loopgp, g -> GroupGroupoidElement( C, root, g ) );
Info( InfoXMod, 1, "(2) loopgp = ", loopgp );
loopgp := GroupWithGenerators( loopgp );
Info( InfoXMod, 1, "(3) loopgp = ", loopgp );
rays := List( rays1, r -> r!.element*eroot );
Info( InfoXMod, 1, "(4) rays = ", rays );
obs := List( rays, g -> ImageElm( h, g ) );
rays := List( rays, r -> GroupGroupoidElement( C, root, r ) );
Info( InfoXMod, 1, "(5) rays = ", rays );
pieces[i] := SinglePieceGroupoidWithRays( loopgp, obs, rays );
od;
gpd := UnionOfPieces( pieces );
SetSize( gpd, size * np );
return gpd;
end );
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