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#
# Pregroup of free groups is special (at the moment at least)
#
InstallGlobalFunction(PregroupOfFreeGroup,
function(n)
local i, r, e, inv, enams;
#T maybe for n <= 26 just use a,b,c,?
enams := [1..2*n+1];
for i in [1..n] do
enams[2 * i] := Concatenation("x", String(i));
enams[2 * i + 1] := Concatenation("X", String(i));
od;
inv := function(x)
if x = 1 then
return 1;
elif IsEvenInt(x) then
return x + 1;
else
return x - 1;
fi;
end;
r := rec( rank := n );
r.fam := NewFamily( "PregroupElementsFamily", IsElementOfPregroup );
r.elt_t := NewType( r!.fam, IsElementOfPregroupOfFreeGroupRep );
r.elts := List( [ 1 .. (2*n + 1) ], i -> Objectify(r.elt_t, rec(parent := r, elt := i)));
r.invs := [];
for e in [1..Length(r.elts)] do
r.elts[e]!.inv := r.elts[inv(e)];
od;
Objectify(PregroupOfFreeGroupType, r);
SetPregroupElementNames(r, enams);
return r;
end);
InstallMethod(ViewString
, "for a pregroup of a free group"
, [IsPregroupOfFreeGroupRep],
function(pg)
return STRINGIFY("<pregroup of free group of rank ", pg!.rank, ">");
end);
InstallMethod(Size
, "for a pregroup of a free group"
, [IsPregroupOfFreeGroupRep],
pg -> 2 * pg!.rank + 1);
InstallMethod(\[\]
, "for a pregroup of a free group"
, [IsPregroupOfFreeGroupRep, IsInt],
function(f,a)
return f!.elts[a];
end);
InstallMethod(Iterator
, "for a pregroup of a free group"
, [IsPregroupOfFreeGroupRep],
function(pgp)
local r;
r := rec( pgp := pgp
, pos := 0
, length := Size(pgp)
, NextIterator := function(iter)
if iter!.pos < iter!.length then
iter!.pos := iter!.pos + 1;
return iter!.pgp[iter!.pos];
else
return fail;
fi;
end
, IsDoneIterator := iter -> iter!.pos = iter!.length
, ShallowCopy := iter -> rec( pgp := iter!.pgp, pos := iter!.pos )
);
return IteratorByFunctions(r);
end);
InstallMethod(PregroupElementNames
, "for a pregroup"
, [IsPregroupOfFreeGroupRep]
, p -> p!.elementnames );
InstallMethod(SetPregroupElementNames
, "for a pregroup"
, [IsPregroupOfFreeGroupRep, IsList]
, function(p, n)
p!.elementnames := n;
end );
InstallMethod(IntermultPairs
, "for a pregroup in table rep"
, [IsPregroupOfFreeGroupRep],
pg -> []);
InstallMethod(IntermultPairsIDs
, "for a pregroup in table rep"
, [IsPregroupOfFreeGroupRep],
pg -> []);
InstallMethod(IntermultMapIDs
, "for a pregroup in table rep"
, [IsPregroupOfFreeGroupRep],
pg -> ListWithIdenticalEntries(Size(pg), []));
InstallMethod(IntermultMap
, "for a pregroup in table rep"
, [IsPregroupOfFreeGroupRep],
pg -> ListWithIdenticalEntries(Size(pg), []));
InstallMethod(One, "for a pregroup",
[IsPregroupOfFreeGroupRep],
pg -> pg!.elts[1]);
#
# PregroupOfFreeGroup elements
#
InstallMethod(IntermultMap
, "for a pregroup element"
, [IsElementOfPregroupOfFreeGroupRep],
pge -> []);
InstallMethod(ViewString
, "for a pregroup element"
, [IsElementOfPregroupOfFreeGroupRep],
function(pge)
if pge!.elt > 0 then
return PregroupElementNames(pge!.parent)[pge!.elt];
else
return "undefined";
fi;
end);
InstallMethod(String
, "for a pregroup element"
, [IsElementOfPregroupOfFreeGroupRep],
function(pge)
if pge!.elt > 0 then
return PregroupElementNames(pge!.parent)[pge!.elt];
else
return "undefined";
fi;
end);
#XXX Is fail as a result for multiplication acceptable?
InstallMethod(\*
, "for pregroup elements"
, IsIdenticalObj
, [IsElementOfPregroupOfFreeGroupRep, IsElementOfPregroupOfFreeGroupRep]
, 0,
function(x,y)
if x!.elt = 1 then
return y;
elif y!.elt = 1 then
return x;
elif x!.inv = y then
return One(PregroupOf(x));
else
return fail;
fi;
end);
InstallMethod(\=
, "for pregroup elements"
, IsIdenticalObj
, [IsElementOfPregroupOfFreeGroupRep, IsElementOfPregroupOfFreeGroupRep]
, 0,
function(x,y)
return x!.elt = y!.elt;
end);
# Artificial ordering on pregroup to make sets work
InstallMethod(\<
, "for pregroup elements"
, IsIdenticalObj
, [ IsElementOfPregroupOfFreeGroupRep, IsElementOfPregroupOfFreeGroupRep]
, 0,
function(x,y)
return x!.elt < y!.elt;
end);
InstallMethod(PregroupOf
, "for pregroup elements"
, [ IsElementOfPregroupOfFreeGroupRep ]
, 0,
function(a)
return a!.parent;
end);
InstallMethod(PregroupInverse
, "for pregroup elements"
, [ IsElementOfPregroupOfFreeGroupRep ]
, 0,
a -> a!.inv);
InstallMethod(PregroupElementId
, "for pregroup elements"
, [ IsElementOfPregroupOfFreeGroupRep ]
, 0,
a -> a!.elt);
InstallMethod(__ID
, "for pregroup elements"
, [ IsElementOfPregroupOfFreeGroupRep ]
, 0,
x -> x!.elt);
InstallMethod(IsDefinedMultiplication
, "for pregroup elements"
, IsIdenticalObj
, [ IsElementOfPregroupOfFreeGroupRep, IsElementOfPregroupOfFreeGroupRep ]
, 0,
function(a,b)
return (a!.elt = 1) or (b!.elt = 1) or (a!.inv = b);
end);
# We could cache intermult pairs,
# or predetermine them, depending
# on the number of intermult lookups
# that could benefit runtime
InstallMethod(IsIntermultPair
, "for pregroup elements"
, IsIdenticalObj
, [IsElementOfPregroupOfFreeGroupRep, IsElementOfPregroupOfFreeGroupRep]
, 0,
function(a,b)
return false;
end);
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
]
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