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############################################################################
##
#W semitran.gi GAP library J. D. Mitchell
##
##
#Y Copyright (C) 2013, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the implementation of some basics for transformation
## semigroups and is based on earlier code of Isabel Araújo and Robert Arthur.
##
InstallMethod(PrintObj,
"for a semigroup with known generators",
[IsTransformationSemigroup and IsGroup and HasGeneratorsOfMagma],
function(S)
Print("Group( ", GeneratorsOfMagma(S), " )");
end);
InstallMethod(SemigroupViewStringPrefix, "for a transformation semigroup",
[IsTransformationSemigroup], S -> "\>transformation\< ");
InstallMethod(SemigroupViewStringSuffix, "for a transformation semigroup",
[IsTransformationSemigroup],
function(S)
return Concatenation("\>degree \>",
ViewString(DegreeOfTransformationSemigroup(S)),
"\<\< ");
end);
InstallMethod(\<, "for transformation semigroups",
[IsTransformationSemigroup, IsTransformationSemigroup],
function(S, T)
return AsSet(S)<AsSet(T);
end);
InstallMethod(MovedPoints, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> MovedPoints(GeneratorsOfSemigroup(s)));
InstallMethod(NrMovedPoints, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> NrMovedPoints(GeneratorsOfSemigroup(s)));
InstallMethod(LargestMovedPoint, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> LargestMovedPoint(GeneratorsOfSemigroup(s)));
InstallMethod(SmallestMovedPoint, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> SmallestMovedPoint(GeneratorsOfSemigroup(s)));
InstallMethod(LargestImageOfMovedPoint, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> LargestImageOfMovedPoint(GeneratorsOfSemigroup(s)));
InstallMethod(SmallestImageOfMovedPoint, "for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
s-> SmallestImageOfMovedPoint(GeneratorsOfSemigroup(s)));
#
InstallMethod(DisplayString, "for a transformation semigroup with generators",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup], ViewString);
#
InstallMethod(ViewString, "for a full transformation semigroup",
[IsTransformationSemigroup and IsFullTransformationSemigroup and
HasGeneratorsOfSemigroup], SUM_FLAGS,
function(S)
return STRINGIFY("<full transformation monoid of degree ",
DegreeOfTransformationSemigroup(S), ">");
end);
InstallMethod(AsMonoid,
"for transformation semigroup with generators",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
function(S)
if MultiplicativeNeutralElement(S) = fail then
return fail;
fi;
return Range(IsomorphismTransformationMonoid(S));
end);
#
InstallTrueMethod(IsFinite, IsTransformationSemigroup);
#
InstallMethod(DegreeOfTransformationSemigroup,
"for a transformation semigroup with generators",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
function(s)
return DegreeOfTransformationCollection(GeneratorsOfSemigroup(s));
end);
#
InstallMethod(DegreeOfTransformationSemigroup,
"for a transformation group with generators",
[IsTransformationSemigroup and HasGeneratorsOfGroup],
function(S)
if not IsEmpty(GeneratorsOfGroup(S)) then
return DegreeOfTransformationCollection(GeneratorsOfGroup(S));
else # What is an example where this can happen?
return DegreeOfTransformationCollection(GeneratorsOfSemigroup(S));
fi;
end);
#
InstallMethod(IsomorphismPermGroup,
"for a group H-class of a semigroup",
[IsGreensHClass],
function( h )
local enum, permgroup, i, perm, j, elts;
if not(IsFinite(h)) then
# What is an example where this can happen?
TryNextMethod();
fi;
if not( IsGroupHClass(h) ) then
# What is an example where this can happen?
ErrorNoReturn("can only create isomorphisms of group H-classes");
fi;
elts:=[];
enum := Enumerator( h );
permgroup:=Group(());
i := 1;
while IsBound( enum[ i ] ) do
perm := [];
j := 1;
while IsBound( enum[ j ] ) do
perm[j]:=Position( enum, enum[j] * enum[ i ] );
j := j+1;
od;
elts[i]:=PermList(perm);
permgroup:=ClosureGroup(permgroup, elts[i]);
i := i+1;
od;
return MappingByFunction( h, permgroup, a -> elts[Position( enum, a )]);
end);
# TODO can this be removed? It doesn't seem to work
InstallMethod(IsomorphismTransformationSemigroup,
"for a semigroup of general mappings",
[IsSemigroup and IsGeneralMappingCollection and HasGeneratorsOfSemigroup],
function( s )
local egens, gens, mapfun;
egens := GeneratorsOfSemigroup(s);
if not ForAll(egens, g->IsMapping(g)) then
return fail;
fi;
gens := List(egens, g->TransformationRepresentation(g)!.transformation);
mapfun := a -> TransformationRepresentation(a)!.transformation;
return MagmaHomomorphismByFunctionNC( s, Semigroup(gens), mapfun );
end);
#
InstallGlobalFunction(FullTransformationSemigroup,
function(d)
local gens, s, i;
if not IsPosInt(d) then
ErrorNoReturn("the argument must be a positive integer");
fi;
if d =1 then
gens:=[Transformation([1])];
elif d=2 then
gens:=[Transformation([2,1]), Transformation([1,1])];
else
gens:=List([1..3], x-> EmptyPlist(d));
gens[1][d]:=1; gens[2][1]:=2; gens[2][2]:=1; gens[3][d]:=1;
for i in [1..d-1] do
gens[1][i]:=i+1;
gens[3][i]:=i;
od;
for i in [3..d] do
gens[2][i]:=i;
od;
Apply(gens, Transformation);
fi;
s:=Monoid(gens);
SetSize(s,d^d);
SetIsFullTransformationSemigroup(s,true);
SetIsRegularSemigroup(s, true);
return s;
end);
#
InstallMethod(IsFullTransformationSemigroup, "for a semigroup",
[IsSemigroup], ReturnFalse);
#
InstallMethod(IsFullTransformationSemigroup, "for a transformation semigroup",
[IsTransformationSemigroup],
function(s)
local n, t;
n := DegreeOfTransformationSemigroup(s);
if n = 0 and HasIsTrivial(s) and IsTrivial(s) then
return true;
elif HasSize(s) then
return Size(s)=n^n;
fi;
t:=FullTransformationSemigroup(DegreeOfTransformationSemigroup(s));
return ForAll(GeneratorsOfSemigroup(t), x-> x in s);
end);
#
InstallMethod(\in,
"for a transformation and a full transformation semigroup",
[IsTransformation, IsFullTransformationSemigroup],
function(e,tn)
return DegreeOfTransformation(e)<=DegreeOfTransformationSemigroup(tn);
end);
#
InstallMethod(Enumerator, "for a full transformation semigroup",
[IsFullTransformationSemigroup], 5,
#to beat the method for an acting semigroup with generators
function(S)
local n, Membership, PrintObj;
n:=DegreeOfTransformationSemigroup(S);
return EnumeratorByFunctions(S, rec(
ElementNumber:=function(enum, pos)
if pos>n^n then
return fail;
fi;
return TransformationNumber(pos, n);
end,
NumberElement:=function(enum, elt)
if DegreeOfTransformation(elt)>n then
return fail;
fi;
return NumberTransformation(elt, n);
end,
Length:=function(enum);
return Size(S);
end,
Membership:=function(elt, enum)
return elt in S;
end,
PrintObj:=function(enum)
Print("<enumerator of full transformation semigroup on ", n," pts>");
end));
end);
# Isomorphism from an arbitrary semigroup to a transformation semigroup, this
# is the fall back method
InstallMethod(IsomorphismTransformationSemigroup, "for a semigroup",
[IsSemigroup],
function(S)
local en, gens, dom, act, pos, T;
en := EnumeratorSorted(S);
if HasGeneratorsOfSemigroup(S) then
gens := GeneratorsOfSemigroup(S);
else
gens := en;
fi;
if HasMultiplicativeNeutralElement(S)
and MultiplicativeNeutralElement(S) <> fail then
dom := en;
act := OnRight;
pos := Position(en, MultiplicativeNeutralElement(S));
else
dom := [1 .. Length(en) + 1];
act := function(i, x)
if i <= Length(en) then
return Position(en, en[i] * x);
fi;
return Position(en, x);
end;
pos := Length(en) + 1;
fi;
T := Semigroup(List(gens, x -> TransformationOp(x, dom, act)));
UseIsomorphismRelation(S, T);
return MagmaIsomorphismByFunctionsNC(S,
T,
x -> TransformationOp(x, dom, act),
x -> en[pos ^ x]);
end);
# Isomorphism from an IsMonoidAsSemigroup to a transformation monoid, this
# is the fall back method
InstallMethod(IsomorphismTransformationMonoid, "for a semigroup",
[IsSemigroup],
function(S)
local iso1, inv1, iso2, inv2;
if MultiplicativeNeutralElement(S) = fail then
ErrorNoReturn("the argument must be a semigroup with a ",
"multiplicative neutral element");
fi;
iso1 := IsomorphismTransformationSemigroup(S);
inv1 := InverseGeneralMapping(iso1);
iso2 := IsomorphismTransformationMonoid(Range(iso1));
inv2 := InverseGeneralMapping(iso2);
UseIsomorphismRelation(S, Range(iso2));
return MagmaIsomorphismByFunctionsNC(S,
Range(iso2),
x -> (x ^ iso1) ^ iso2,
x -> (x ^ inv2) ^ inv1);
end);
# Isomorphism from an IsMonoidAsSemigroup transformation semigroup to a
# transformation monoid
InstallMethod(IsomorphismTransformationMonoid,
"for a transformation semigroup",
[IsTransformationSemigroup and HasGeneratorsOfSemigroup],
function(S)
local id, dom, T, inv;
if IsMonoid(S) then
return MappingByFunction(S, S, IdFunc, IdFunc);
fi;
if MultiplicativeNeutralElement(S) = fail then
ErrorNoReturn("the argument must be a semigroup with a ",
"multiplicative neutral element");
fi;
id := MultiplicativeNeutralElement(S);
dom := ImageSetOfTransformation(id, DegreeOfTransformationSemigroup(S));
T := Monoid(List(GeneratorsOfSemigroup(S),
x -> TransformationOp(x, dom)));
UseIsomorphismRelation(S, T);
inv := function(x)
local out, i;
out := [1 .. DegreeOfTransformationSemigroup(S)];
for i in [1 .. Length(dom)] do
out[dom[i]] := dom[i ^ x];
od;
return id * Transformation(out);
end;
return MagmaIsomorphismByFunctionsNC(S,
T,
x -> TransformationOp(x, dom),
inv);
end);
InstallMethod(IsomorphismTransformationSemigroup,
"for a transformation semigroup",
[IsTransformationSemigroup],
function(S)
return MagmaIsomorphismByFunctionsNC(S, S, IdFunc, IdFunc);
end);
InstallMethod(IsomorphismTransformationSemigroup, "for partial perm semigroup",
[IsPartialPermSemigroup],
function(S)
local n, T, inv;
n := Maximum(DegreeOfPartialPermCollection(S),
CodegreeOfPartialPermCollection(S)) + 1;
T := Semigroup(List(GeneratorsOfSemigroup(S), x -> AsTransformation(x, n)));
UseIsomorphismRelation(S, T);
inv := function(x)
local out, j, i;
out := [];
for i in [1 .. n - 1] do
j := i ^ x;
if j <> n then
out[i] := j;
else
out[i] := 0;
fi;
od;
return PartialPerm(out);
end;
return MagmaIsomorphismByFunctionsNC(S,
T,
x -> AsTransformation(x, n),
inv);
end);
InstallMethod(IsomorphismTransformationMonoid, "for partial perm semigroup",
[IsPartialPermSemigroup],
function(S)
local n, T, inv;
if not (IsMonoid(S) or One(S) <> fail) then
ErrorNoReturn("the argument must be a semigroup with a ",
"multiplicative neutral element");
# in the case of partial perm semigroups having a One is the equivalent to
# having a MultiplicativeNeutralElement
fi;
n := Maximum(DegreeOfPartialPermCollection(S),
CodegreeOfPartialPermCollection(S)) + 1;
T := Monoid(List(GeneratorsOfSemigroup(S), x -> AsTransformation(x, n)));
UseIsomorphismRelation(S, T);
inv := function(x)
local out, j, i;
out := [];
for i in [1 .. n - 1] do
j := i ^ x;
if j <> n then
out[i] := j;
else
out[i] := 0;
fi;
od;
return PartialPerm(out);
end;
return MagmaIsomorphismByFunctionsNC(S,
T,
x -> AsTransformation(x, n),
inv);
end);
# Isomorphisms from perm groups to transformation semigroups/monoids
InstallMethod(IsomorphismTransformationMonoid,
"for a perm group with generators",
[IsPermGroup and HasGeneratorsOfGroup],
function(G)
local S;
S := Monoid(List(GeneratorsOfGroup(G), AsTransformation));
UseIsomorphismRelation(G, S);
SetIsGroupAsSemigroup(S, true);
return MagmaIsomorphismByFunctionsNC(G, S, AsTransformation, AsPermutation);
end);
InstallMethod(IsomorphismTransformationSemigroup,
"for a perm group with generators",
[IsPermGroup and HasGeneratorsOfGroup],
function(G)
local S;
# The next line has to use Semigroup instead of Monoid so that S has the
# correct set of generators
S := Semigroup(List(GeneratorsOfGroup(G), AsTransformation));
UseIsomorphismRelation(G, S);
SetIsGroupAsSemigroup(S, true);
return MagmaIsomorphismByFunctionsNC(G, S, AsTransformation, AsPermutation);
end);
InstallMethod(AntiIsomorphismTransformationSemigroup, "for a semigroup",
[IsSemigroup],
function(S)
local en, gens, dom, act, pos, T;
en := EnumeratorSorted(S);
if HasGeneratorsOfSemigroup(S) then
gens := GeneratorsOfSemigroup(S);
else
gens := en;
fi;
if HasMultiplicativeNeutralElement(S)
and MultiplicativeNeutralElement(S) <> fail then
dom := en;
act := function(pt, x)
return x * pt;
end;
pos := Position(en, MultiplicativeNeutralElement(S));
else
dom := [1 .. Length(en) + 1];
act := function(i, x)
if i <= Length(en) then
return Position(en, x * en[i]);
fi;
return Position(en, x);
end;
pos := Length(en) + 1;
fi;
T := Semigroup(List(gens, x -> TransformationOp(x, dom, act)));
return MagmaIsomorphismByFunctionsNC(S,
T,
x -> TransformationOp(x, dom, act),
x -> en[pos ^ x]);
end);