Quelle gpgpd.gi Sprache: unbekannt

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#W  gpgpd.gi                   GAP4 package `XMod'              Chris Wensley
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#Y  Copyright (C) 2001-2024, Chris Wensley et al,

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#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 );

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#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 );

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#M  \*( <e1> <e2> ) . . . . . . . . . . . . . for two group-groupoid elements
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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 );

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#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 );

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#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 );

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#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 );

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#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 );

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#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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