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#############################################################################
##
#W reesmat.gi GAP library J. D. Mitchell
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2013 The GAP Group
##
## This file contains the implementation of Rees matrix semigroups.
##
# Notes: there are essentially 3 types of semigroups here:
# 1) the whole family Rees matrix semigroup (notice that the matrix used to
# define the semigroup may contain rows or columns consisting entirely of 0s
# so it is not guarenteed that the resulting semigroup is 0-simple)
# 2) subsemigroups U of 1), which defined by a generating set and are Rees matrix
# semigroups, i.e. U=I'xG'xJ' where the whole family if IxGxJ and I', J' subsets
# of I, J and G' a subsemigroup of G.
# 3) subsemigroups of 1) obtained by removing an index or element of the
# underlying semigroup, and hence are also Rees matrix semigroups.
# 4) subsemigroups of 1) defined by generating sets which are not
# simple/0-simple.
# So, the methods with filters IsRees(Zero)MatrixSemigroup and
# HasGeneratorsOfSemigroup only apply to subsemigroups of type 2).
# Subsemigroups of type 3 already know the values of Rows, Columns,
# UnderlyingSemigroup, and Matrix.
# a Rees matrix semigroup over a semigroup <S> is simple if and only if <S> is
# simple.
InstallImmediateMethod(IsFinite, IsReesZeroMatrixSubsemigroup, 0,
function(R)
if IsBound(ElementsFamily(FamilyObj(R))!.IsFinite) then
if HasIsWholeFamily(R) and IsWholeFamily(R) then
return ElementsFamily(FamilyObj(R))!.IsFinite;
elif ElementsFamily(FamilyObj(R))!.IsFinite then
return true;
fi;
fi;
TryNextMethod();
end);
InstallImmediateMethod(IsFinite, IsReesMatrixSubsemigroup, 0,
function(R)
if IsBound(ElementsFamily(FamilyObj(R))!.IsFinite) then
if HasIsWholeFamily(R) and IsWholeFamily(R) then
return ElementsFamily(FamilyObj(R))!.IsFinite;
elif ElementsFamily(FamilyObj(R))!.IsFinite then
return true;
fi;
fi;
TryNextMethod();
end);
InstallMethod(IsFinite, "for a Rees 0-matrix subsemigroup",
[IsReesZeroMatrixSubsemigroup],
function(R)
if HasIsFinite(ParentAttr(R)) and IsFinite(ParentAttr(R))then
return true;
fi;
TryNextMethod();
end);
InstallMethod(IsFinite, "for a Rees matrix subsemigroup",
[IsReesMatrixSubsemigroup],
function(R)
if HasIsFinite(ParentAttr(R)) and IsFinite(ParentAttr(R)) then
return true;
fi;
TryNextMethod();
end);
#
InstallMethod(IsIdempotent, "for a Rees 0-matrix semigroup element",
[IsReesZeroMatrixSemigroupElement],
function(x)
local matrix_entry, g;
if x![1] = 0 then
# <x> is the 0 element of the family
return true;
fi;
matrix_entry := x![4][x![3]][x![1]];
if matrix_entry = 0 then
return false;
fi;
g := UnderlyingElementOfReesZeroMatrixSemigroupElement(x);
return g * matrix_entry * g = g;
end);
#
InstallTrueMethod(IsRegularSemigroup,
IsReesMatrixSemigroup and IsSimpleSemigroup);
#
InstallMethod(IsRegularSemigroup, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup],
function(R)
local mat;
mat := Matrix(R);
if HasIsGroupAsSemigroup(UnderlyingSemigroup(R))
and IsGroupAsSemigroup(UnderlyingSemigroup(R)) then
return ForAll(Rows(R), i -> ForAny(Columns(R), j -> mat[j][i]<>0))
and ForAll(Columns(R), j -> ForAny(Rows(R), i -> mat[j][i]<>0));
else
TryNextMethod();
fi;
end);
#
InstallMethod(IsSimpleSemigroup,
"for a subsemigroup of a Rees matrix semigroup with an underlying semigroup",
[IsReesMatrixSubsemigroup and HasUnderlyingSemigroup],
R-> IsSimpleSemigroup(UnderlyingSemigroup(R)));
# check that the matrix has no rows or columns consisting entirely of 0s
# and that the underlying semigroup is simple
InstallMethod(IsZeroSimpleSemigroup, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup],
function(R)
local i, mat;
mat := Matrix(R);
if ForAny(Columns(R), j -> ForAll(Rows(R), i -> mat[j][i] = 0))
or ForAny(Rows(R), i -> ForAll(Columns(R), j -> mat[j][i] = 0)) then
return false;
fi;
return IsSimpleSemigroup(UnderlyingSemigroup(R));
end);
#
InstallMethod(IsReesMatrixSemigroup, "for a semigroup", [IsSemigroup], ReturnFalse);
#
InstallMethod(IsReesMatrixSemigroup,
"for a Rees matrix subsemigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local gens, I, J;
if IsWholeFamily(R) then
return true;
fi;
# it is still possible that <R> is a Rees matrix semigroup, if, for example,
# we have a subsemigroup specified by generators which equals a subsemigroup
# obtained by removing a row, in the case that <R> is not simple.
gens:=GeneratorsOfSemigroup(R);
I:=Set(List(gens, x-> x![1]));
J:=Set(List(gens, x-> x![3]));
return ForAll(GeneratorsOfReesMatrixSemigroupNC(ParentAttr(R), I,
Semigroup(List(AsSSortedList(R), x-> x![2])), J), x-> x in R);
end);
#
InstallMethod(IsReesZeroMatrixSemigroup, "for a semigroup", [IsSemigroup],
ReturnFalse);
#
InstallMethod(IsReesZeroMatrixSemigroup,
"for a Rees 0-matrix subsemigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local gens, pos, elts, I, J;
if IsWholeFamily(R) then
return true;
fi;
# it is still possible that <R> is a Rees 0-matrix semigroup, if, for
# example, we have a subsemigroup specified by generators which equals a
# subsemigroup obtained by removing a row, in the case that <R> is not simple.
if MultiplicativeZero(R)<>MultiplicativeZero(ParentAttr(R)) then
return false; #Rees 0-matrix semigroups always contain the 0.
fi;
gens := Unique(GeneratorsOfSemigroup(R));
pos:=Position(gens, MultiplicativeZero(R));
if pos<>fail then
if Size(gens) = 1 then
return false;
fi;
gens:=ShallowCopy(gens);
Remove(gens, pos);
fi;
elts:=ShallowCopy(AsSSortedList(R));
RemoveSet(elts, MultiplicativeZero(R));
I:=Set(List(gens, x-> x![1]));
J:=Set(List(gens, x-> x![3]));
return ForAll(GeneratorsOfReesZeroMatrixSemigroupNC(ParentAttr(R), I,
Semigroup(List(elts, x-> x![2])), J), x-> x in R);
end);
#
InstallMethod(ReesMatrixSemigroup, "for a semigroup and a rectangular table",
[IsSemigroup, IsRectangularTable],
function(S, mat)
local fam, R, type, x;
for x in mat do
if ForAny(x, s-> not s in S) then
ErrorNoReturn("the entries of the second argument (a rectangular ",
"table) must belong to the first argument (a semigroup)");
fi;
od;
fam := NewFamily( "ReesMatrixSemigroupElementsFamily",
IsReesMatrixSemigroupElement);
if HasIsFinite(S) then
fam!.IsFinite := IsFinite(S);
fi;
# create the Rees matrix semigroup
R := Objectify( NewType( CollectionsFamily( fam ), IsWholeFamily and
IsReesMatrixSubsemigroup and IsAttributeStoringRep ), rec() );
# store the type of the elements in the semigroup
type:=NewType(fam, IsReesMatrixSemigroupElement);
fam!.type:=type;
SetTypeReesMatrixSemigroupElements(R, type);
SetReesMatrixSemigroupOfFamily(fam, R);
SetMatrix(R, mat); SetUnderlyingSemigroup(R, S);
SetRows(R, [1..Length(mat[1])]);
SetColumns(R, [1..Length(mat)]);
if HasIsSimpleSemigroup(S) and IsSimpleSemigroup(S) then
SetIsSimpleSemigroup(R, true);
fi;
if HasIsFinite(S) then
SetIsFinite(R, IsFinite(S));
fi;
SetIsZeroSimpleSemigroup(R, false);
return R;
end);
#
InstallMethod(ReesZeroMatrixSemigroup, "for a semigroup and a dense list",
[IsSemigroup, IsDenseList],
function(S, mat)
local fam, R, type, x;
if IsEmpty(mat) or not ForAll(mat, x -> IsDenseList(x) and not IsEmpty(x)
and Length(x) = Length(mat[1])) then
ErrorNoReturn("the second argument must be a non-empty list, whose ",
"entries are non-empty lists of equal length");
fi;
for x in mat do
if ForAny(x, s -> not (s = 0 or s in S)) then
ErrorNoReturn("the entries of the second argument must be 0 or belong ",
"to the first argument (a semigroup)");
fi;
od;
fam := NewFamily("ReesZeroMatrixSemigroupElementsFamily",
IsReesZeroMatrixSemigroupElement);
if HasIsFinite(S) then
fam!.IsFinite := IsFinite(S);
fi;
# create the Rees matrix semigroup
R := Objectify(NewType(CollectionsFamily(fam),
IsWholeFamily
and IsReesZeroMatrixSubsemigroup
and IsAttributeStoringRep), rec());
# store the type of the elements in the semigroup
type := NewType(fam, IsReesZeroMatrixSemigroupElement);
fam!.type := type;
SetTypeReesMatrixSemigroupElements(R, type);
SetReesMatrixSemigroupOfFamily(fam, R);
SetMatrix(R, mat);
SetUnderlyingSemigroup(R, S);
SetRows(R, [1 .. Length(mat[1])]);
SetColumns(R, [1 .. Length(mat)]);
SetMultiplicativeZero(R,
Objectify(TypeReesMatrixSemigroupElements(R), [0]));
# cannot set IsZeroSimpleSemigroup to be <true> here since the matrix may
# contain a row or column consisting entirely of 0s!
# WW Also S might not be a simple semigroup (which is necessary)!
if IsGroup(S) or (HasIsFinite(S) and IsFinite(S)) then
GeneratorsOfSemigroup(R);
fi;
SetIsSimpleSemigroup(R, false);
return R;
end);
InstallMethod(ViewObj, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup], 3, #to beat the next method
function(R)
Print("\>\><Rees matrix semigroup \>", Length(Rows(R)), "x",
Length(Columns(R)), "\< over \<");
View(UnderlyingSemigroup(R));
Print(">\<");
return;
end);
#
InstallMethod(ViewObj, "for a subsemigroup of a Rees matrix semigroup",
[IsReesMatrixSubsemigroup], PrintObj);
#
InstallMethod(PrintObj, "for a subsemigroup of a Rees matrix semigroup",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
Print("\><subsemigroup of \>", Length(Rows(ParentAttr(R))), "x",
Length(Columns(ParentAttr(R))), "\< Rees matrix semigroup \>with ");
Print(Length(GeneratorsOfSemigroup(R)));
Print(" generator");
if Length(GeneratorsOfSemigroup(R))>1 then
Print("s");
fi;
Print("\<>\<");
return;
end);
#
InstallMethod(ViewObj, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup], 3, #to beat the next method
function(R)
Print("\>\><Rees 0-matrix semigroup \>", Length(Rows(R)), "x",
Length(Columns(R)), "\< over \<");
View(UnderlyingSemigroup(R));
Print(">\<");
return;
end);
#
InstallMethod(ViewObj, "for a subsemigroup of a Rees 0-matrix semigroup",
[IsReesZeroMatrixSubsemigroup], PrintObj);
InstallMethod(PrintObj, "for a subsemigroup of a Rees 0-matrix semigroup",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
Print("\><subsemigroup of \>", Length(Rows(ParentAttr(R))), "x",
Length(Columns(ParentAttr(R))), "\< Rees 0-matrix semigroup \>with ");
Print(Length(GeneratorsOfSemigroup(R)));
Print(" generator");
if Length(GeneratorsOfSemigroup(R))>1 then
Print("s");
fi;
Print("\<>\<");
return;
end);
#
InstallMethod(PrintObj, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup and IsWholeFamily], 2,
function(R)
Print("ReesMatrixSemigroup( ");
Print(UnderlyingSemigroup(R));
Print(", ");
Print(Matrix(R));
Print(" )");
end);
#
InstallMethod(PrintObj, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup and IsWholeFamily], 2,
function(R)
Print("ReesZeroMatrixSemigroup( ");
Print(UnderlyingSemigroup(R));
Print(", ");
Print(Matrix(R));
Print(" )");
end);
#
InstallMethod(Size, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup],
function(R)
if Size(UnderlyingSemigroup(R))=infinity then
return infinity;
fi;
return Length(Rows(R))*Size(UnderlyingSemigroup(R))*Length(Columns(R));
end);
#
InstallMethod(Size, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup],
function(R)
if Size(UnderlyingSemigroup(R))=infinity then
return infinity;
fi;
return Length(Rows(R))*Size(UnderlyingSemigroup(R))*Length(Columns(R))+1;
end);
#
InstallMethod(Representative,
"for a subsemigroup of Rees matrix semigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup], 2,
# to beat the other method
function(R)
return GeneratorsOfSemigroup(R)[1];
end);
#
InstallMethod(Representative, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup],
function(R)
return Objectify(TypeReesMatrixSemigroupElements(R),
[Rows(R)[1], Representative(UnderlyingSemigroup(R)), Columns(R)[1],
Matrix(R)]);
end);
#
InstallMethod(Representative,
"for a subsemigroup of Rees 0-matrix semigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup], 2,
# to beat the other method
function(R)
return GeneratorsOfSemigroup(R)[1];
end);
#
InstallMethod(Representative, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup],
function(R)
return Objectify(TypeReesMatrixSemigroupElements(R),
[Rows(R)[1], Representative(UnderlyingSemigroup(R)), Columns(R)[1],
Matrix(R)]);
end);
#
InstallMethod(Enumerator, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup],
function( R )
return EnumeratorByFunctions(R, rec(
enum:=EnumeratorOfCartesianProduct(Rows(R),
Enumerator(UnderlyingSemigroup(R)), Columns(R), [Matrix(R)]),
NumberElement:=function(enum, x)
if FamilyObj(x) <> ElementsFamily(FamilyObj(R)) then
return fail;
fi;
return Position(enum!.enum, [x![1], x![2], x![3], x![4]]);
end,
ElementNumber:=function(enum, n)
return Objectify(TypeReesMatrixSemigroupElements(R), enum!.enum[n]);
end,
Length:=enum-> Length(enum!.enum),
PrintObj:=function(enum)
Print("<enumerator of Rees matrix semigroup>");
return;
end));
end);
#
InstallMethod(Enumerator, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup and HasUnderlyingSemigroup],
function( R )
return EnumeratorByFunctions(R, rec(
enum:=EnumeratorOfCartesianProduct(Rows(R),
Enumerator(UnderlyingSemigroup(R)), Columns(R), [Matrix(R)]),
NumberElement:=function(enum, x)
local pos;
if FamilyObj(x) <> ElementsFamily(FamilyObj(R)) then
return fail;
elif IsMultiplicativeZero(R, x) then
return 1;
fi;
pos:=Position(enum!.enum, [x![1], x![2], x![3], x![4]]);
if pos=fail then
return fail;
fi;
return pos+1;
end,
ElementNumber:=function(enum, n)
if n=1 then
return MultiplicativeZero(R);
fi;
return Objectify(TypeReesMatrixSemigroupElements(R), enum!.enum[n-1]);
end,
Length:=enum-> Length(enum!.enum)+1,
PrintObj:=function(enum)
Print("<enumerator of Rees 0-matrix semigroup>");
return;
end));
end);
# these methods (Matrix, Rows, Columns, UnderlyingSemigroup) should only apply
# to subsemigroups defined by a generating set, which happen to be
# simple/0-simple.
InstallMethod(Matrix, "for a Rees matrix semigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
if not IsReesMatrixSemigroup(R) then
return fail;
fi;
return Matrix(ParentAttr(R));
end);
InstallMethod(Matrix, "for a Rees 0-matrix semigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
if not IsReesZeroMatrixSemigroup(R) then
return fail;
fi;
return Matrix(ParentAttr(R));
end);
#
InstallMethod(Rows, "for a Rees matrix semigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
if not IsReesMatrixSemigroup(R) then
return fail;
fi;
return SetX(GeneratorsOfSemigroup(R), x-> x![1]);
end);
InstallMethod(Rows, "for a Rees 0-matrix semigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local out;
if not IsReesZeroMatrixSemigroup(R) then
return fail;
fi;
out:=SetX(GeneratorsOfSemigroup(R), x-> x![1]);
if out[1]=0 then
Remove(out, 1);
fi;
return out;
end);
#
InstallMethod(Columns, "for a Rees matrix semigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
if not IsReesMatrixSemigroup(R) then
return fail;
fi;
return SetX(GeneratorsOfSemigroup(R), x-> x![3]);
end);
InstallMethod(Columns, "for a Rees 0-matrix semigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local out, x;
if not IsReesZeroMatrixSemigroup(R) then
return fail;
fi;
out:=[];
for x in GeneratorsOfSemigroup(R) do
if x![1]<>0 then
AddSet(out, x![3]);
fi;
od;
return out;
end);
# these methods only apply to subsemigroups which happen to be Rees matrix
# semigroups
InstallMethod(UnderlyingSemigroup,
"for a Rees matrix semigroup with generators",
[IsReesMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local gens, i, S, U;
if not IsReesMatrixSemigroup(R) then
return fail;
fi;
gens:=List(AsSSortedList(R), x-> x![2]);
if IsGeneratorsOfMagmaWithInverses(gens) then
i:=1;
S:=UnderlyingSemigroup(ParentAttr(R));
U:=Group(gens[1]);
while Size(U)<Size(S) and i<Length(gens) do
i:=i+1;
U:=ClosureGroup(U, gens[i]);
od;
else
U:=Semigroup(gens);
fi;
SetIsSimpleSemigroup(U, true);
return U;
end);
# these methods only apply to subsemigroups which happen to be Rees matrix
# semigroups
InstallMethod(UnderlyingSemigroup,
"for a Rees 0-matrix semigroup with generators",
[IsReesZeroMatrixSubsemigroup and HasGeneratorsOfSemigroup],
function(R)
local gens, i, S, U;
if not IsReesZeroMatrixSemigroup(R) then
return fail;
fi;
#remove the 0
gens:=Filtered(AsSSortedList(R), x-> x![1]<>0);
Apply(gens, x-> x![2]);
gens := Set(gens);
if IsGeneratorsOfMagmaWithInverses(gens) then
i:=1;
S:=UnderlyingSemigroup(ParentAttr(R));
U:=Group(gens[1]);
while Size(U)<Size(S) and i<Length(gens) do
i:=i+1;
U:=ClosureGroup(U, gens[i]);
od;
else
U:=Semigroup(gens);
fi;
return U;
end);
# again only for subsemigroups defined by generators...
InstallMethod(TypeReesMatrixSemigroupElements,
"for a subsemigroup of Rees matrix semigroup",
[IsReesMatrixSubsemigroup],
R -> TypeReesMatrixSemigroupElements(ParentAttr(R)));
InstallMethod(TypeReesMatrixSemigroupElements,
"for a subsemigroup of Rees 0-matrix semigroup",
[IsReesZeroMatrixSubsemigroup],
R -> TypeReesMatrixSemigroupElements(ParentAttr(R)));
# Elements...
InstallGlobalFunction(RMSElement,
function(R, i, s, j)
local out;
if not (IsReesMatrixSubsemigroup(R)
or IsReesZeroMatrixSubsemigroup(R)) then
ErrorNoReturn("the first argument must be a Rees matrix semigroup",
" or Rees 0-matrix semigroup");
fi;
if (HasIsReesMatrixSemigroup(R) and IsReesMatrixSemigroup(R)) or
(HasIsReesZeroMatrixSemigroup(R) and IsReesZeroMatrixSemigroup(R)) then
if not i in Rows(R) then
ErrorNoReturn("the second argument (a positive integer) does not ",
"belong to the rows of the first argument (a Rees ",
"(0-)matrix semigroup)");
elif not j in Columns(R) then
ErrorNoReturn("the fourth argument (a positive integer) does not ",
"belong to the columns of the first argument (a Rees ",
"(0-)matrix semigroup)");
elif not s in UnderlyingSemigroup(R) then
ErrorNoReturn("the second argument does not belong to the",
"underlying semigroup of the first argument (a Rees ",
"(0-)matrix semgiroup)");
fi;
return Objectify(TypeReesMatrixSemigroupElements(R), [i, s, j, Matrix(R)]);
fi;
out:=Objectify(TypeReesMatrixSemigroupElements(R),
[i, s, j, Matrix(ParentAttr(R))]);
if not out in R then # for the case R is defined by a generating set
ErrorNoReturn("the arguments do not describe an element of the first ",
"argument (a Rees (0-)matrix semigroup)");
fi;
return out;
end);
#
InstallMethod(RowOfReesMatrixSemigroupElement,
"for a Rees matrix semigroup element", [IsReesMatrixSemigroupElement],
x-> x![1]);
#
InstallMethod(RowOfReesZeroMatrixSemigroupElement,
"for a Rees 0-matrix semigroup element", [IsReesZeroMatrixSemigroupElement],
function(x)
if x![1] = 0 then
return fail;
fi;
return x![1];
end);
#
InstallMethod(UnderlyingElementOfReesMatrixSemigroupElement,
"for a Rees matrix semigroup element", [IsReesMatrixSemigroupElement],
x-> x![2]);
#
InstallMethod(UnderlyingElementOfReesZeroMatrixSemigroupElement,
"for a Rees 0-matrix semigroup element", [IsReesZeroMatrixSemigroupElement],
function(x)
if x![1] = 0 then
return fail;
fi;
return x![2];
end);
#
InstallMethod(ColumnOfReesMatrixSemigroupElement,
"for a Rees matrix semigroup element", [IsReesMatrixSemigroupElement],
x-> x![3]);
#
InstallMethod(ColumnOfReesZeroMatrixSemigroupElement,
"for a Rees 0-matrix semigroup element", [IsReesZeroMatrixSemigroupElement],
function(x)
if x![1] = 0 then
return fail;
fi;
return x![3];
end);
#
InstallMethod(PrintObj, "for a Rees matrix semigroup element",
[IsReesMatrixSemigroupElement],
function(x)
Print("RMSElement(",
ReesMatrixSemigroupOfFamily(FamilyObj(x)),
", ", x![1], ", ", x![2], ", ", x![3], ")");
end);
#
InstallMethod(PrintObj, "for a Rees 0-matrix semigroup element",
[IsReesZeroMatrixSemigroupElement],
function(x)
if x![1]=0 then
Print("MultiplicativeZero(",
ReesMatrixSemigroupOfFamily(FamilyObj(x)),
")");
return;
fi;
Print("RMSElement(",
ReesMatrixSemigroupOfFamily(FamilyObj(x)),
", ", x![1], ", ", x![2], ", ", x![3], ")");
end);
InstallMethod(ViewString, "for a Rees matrix semigroup element",
[IsReesMatrixSemigroupElement],
function(x)
return PRINT_STRINGIFY("\>(", ViewString(x![1]), ",", ViewString(x![2]), ",",
ViewString(x![3]), ")\<");
end);
#
InstallMethod(ViewString, "for a Rees 0-matrix semigroup element",
[IsReesZeroMatrixSemigroupElement],
function(x)
if x![1] = 0 then
return "0";
fi;
return PRINT_STRINGIFY("\>(", ViewString(x![1]), ",", ViewString(x![2]), ",",
ViewString(x![3]), ")\<");
end);
#
InstallMethod(ELM_LIST, "for a Rees matrix semigroup element",
[IsReesMatrixSemigroupElement, IsPosInt],
function(x, i)
if i > 3 then
ErrorNoReturn("the second argument must be 1, 2, or 3");
fi;
return x![i];
end);
#
InstallMethod(ELM_LIST, "for a Rees 0-matrix semigroup element",
[IsReesZeroMatrixSemigroupElement, IsPosInt],
function(x, i)
if x![1] = 0 then
ErrorNoReturn("the first argument (an element of a Rees 0-matrix ",
"semigroup) must be non-zero");
elif i > 3 then
ErrorNoReturn("the second argument must be 1, 2, or 3");
fi;
return x![i];
end);
#
InstallMethod(ZeroOp, "for a Rees matrix semigroup element",
[IsReesMatrixSemigroupElement], ReturnFail);
#
InstallMethod(ZeroOp, "for a Rees 0-matrix semigroup element",
[IsReesZeroMatrixSemigroupElement],
function(x)
return MultiplicativeZero(ReesMatrixSemigroupOfFamily(FamilyObj(x)));
end);
#
InstallMethod(MultiplicativeZeroOp, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup], ReturnFail);
#
InstallMethod(\*, "for elements of a Rees matrix semigroup",
IsIdenticalObj,
[IsReesMatrixSemigroupElement, IsReesMatrixSemigroupElement],
function(x, y)
return Objectify(FamilyObj(x)!.type,
[x![1], x![2]*x![4][x![3]][y![1]]*y![2], y![3], x![4]]);
end);
#
InstallMethod(\*, "for elements of a Rees 0-matrix semigroup",
IsIdenticalObj,
[IsReesZeroMatrixSemigroupElement, IsReesZeroMatrixSemigroupElement],
function(x, y)
local p;
if x![1]=0 then
return x;
elif y![1]=0 then
return y;
fi;
p:=x![4][x![3]][y![1]];
if p=0 then
return Objectify(FamilyObj(x)!.type, [0]);
fi;
return Objectify(FamilyObj(x)!.type, [x![1], x![2]*p*y![2], y![3], x![4]]);
end);
#
InstallMethod(\<, "for elements of a Rees matrix semigroup",
IsIdenticalObj,
[IsReesMatrixSemigroupElement, IsReesMatrixSemigroupElement],
function(x, y)
return x![1]<y![1] or (x![1]=y![1] and x![2]<y![2])
or (x![1]=y![1] and x![2]=y![2] and x![3]<y![3]);
end);
# 0 is less than everything!
InstallMethod(\<, "for elements of a Rees 0-matrix semigroup",
IsIdenticalObj,
[IsReesZeroMatrixSemigroupElement, IsReesZeroMatrixSemigroupElement],
function(x, y)
if x![1]=0 then
return y![1]<>0;
elif y![1]=0 then
return false;
fi;
return x![1]<y![1] or (x![1]=y![1] and x![2]<y![2])
or (x![1]=y![1] and x![2]=y![2] and x![3]<y![3]);
end);
#
InstallMethod(\=, "for elements of a Rees matrix semigroup",
IsIdenticalObj,
[IsReesMatrixSemigroupElement, IsReesMatrixSemigroupElement],
function(x, y)
return x![1]=y![1] and x![2]=y![2] and x![3]=y![3];
end);
#
InstallMethod(\=, "for elements of a Rees 0-matrix semigroup",
IsIdenticalObj,
[IsReesZeroMatrixSemigroupElement, IsReesZeroMatrixSemigroupElement],
function(x, y)
if x![1]=0 then
return y![1]=0;
fi;
return x![1]=y![1] and x![2]=y![2] and x![3]=y![3];
end);
#
InstallMethod(ParentAttr, "for a subsemigroup of a Rees matrix semigroup",
[IsReesMatrixSubsemigroup],
function(R)
return ReesMatrixSemigroupOfFamily(FamilyObj(Representative(R)));
end);
#
InstallMethod(ParentAttr, "for a subsemigroup of a Rees 0-matrix semigroup",
[IsReesZeroMatrixSubsemigroup],
function(R)
return ReesMatrixSemigroupOfFamily(FamilyObj(Representative(R)));
end);
#
InstallMethod(IsWholeFamily, "for a subsemigroup of a Rees matrix semigroup",
[IsReesMatrixSubsemigroup],
function(R)
local S;
if Size(R)=Size(ReesMatrixSemigroupOfFamily(ElementsFamily(FamilyObj(R))))
then
if not HasMatrix(R) then # <R> is defined by generators
S:=ParentAttr(R);
SetMatrix(R, Matrix(S));
SetUnderlyingSemigroup(R, UnderlyingSemigroup(S));
SetRows(R, Rows(S));
SetColumns(R, Columns(S));
fi;
return true;
else
return false;
fi;
end);
#
InstallMethod(IsWholeFamily, "for a subsemigroup of a Rees 0-matrix semigroup",
[IsReesZeroMatrixSubsemigroup],
function(R)
local S;
if Size(R)=Size(ReesMatrixSemigroupOfFamily(ElementsFamily(FamilyObj(R))))
then
if not HasMatrix(R) then # <R> is defined by generators
S:=ParentAttr(R);
SetMatrix(R, Matrix(S));
SetUnderlyingSemigroup(R, UnderlyingSemigroup(S));
SetRows(R, Rows(S));
SetColumns(R, Columns(S));
fi;
return true;
else
return false;
fi;
end);
# generators for the subsemigroup generated by <IxUxJ>, when <R> is a Rees
# matrix semigroup (and not only a subsemigroup).
InstallGlobalFunction(GeneratorsOfReesMatrixSemigroupNC,
function(R, I, U, J)
local P, type, i, j, gens, u;
P:=Matrix(R); type:=TypeReesMatrixSemigroupElements(R);
if IsGroup(U) then
i:=I[1]; j:=J[1];
if IsTrivial(U) then
gens:=[Objectify(type, [i, P[j][i]^-1, j, P])];
else
gens:=List(GeneratorsOfGroup(U), x->
Objectify( type, [i, x*P[j][i]^-1, j, P]));
fi;
if Length(I)>Length(J) then
for i in [2..Length(J)] do
Add(gens, Objectify(type, [I[i], One(U), J[i], P]));
od;
for i in [Length(J)+1..Length(I)] do
Add(gens, Objectify(type, [I[i], One(U), J[1], P]));
od;
else
for i in [2..Length(I)] do
Add(gens, Objectify(type, [I[i], One(U), J[i], P]));
od;
for i in [Length(I)+1..Length(J)] do
Add(gens, Objectify(type, [I[1], One(U), J[i], P]));
od;
fi;
else
gens:=[];
for i in I do
for u in U do
for j in J do
Add(gens, Objectify(type, [i, u, j, P]));
od;
od;
od;
fi;
return gens;
end);
# generators for the subsemigroup generated by <IxUxJ>, when <R> is a Rees
# matrix semigroup (and not only a subsemigroup).
InstallGlobalFunction(GeneratorsOfReesZeroMatrixSemigroupNC,
function(R, I, U, J)
local P, type, i, j, gens, k, u;
P:=Matrix(R); type:=TypeReesMatrixSemigroupElements(R);
i:=I[1]; j:=First(J, j-> P[j][i]<>0);
if IsGroup(U) and IsRegularSemigroup(R) and not j=fail then
if IsTrivial(U) then
gens:=[Objectify(type, [i, P[j][i]^-1, j, P])];
else
gens:=List(GeneratorsOfGroup(U), x->
Objectify(type, [i, x*P[j][i]^-1, j, P]));
fi;
for k in J do
if k<>j then
Add(gens, Objectify(type, [i, One(U), k, P]));
fi;
od;
for k in I do
if k<>i then
Add(gens, Objectify(type, [k, One(U), j, P]));
fi;
od;
else
gens:=[];
for i in I do
for u in U do
for j in J do
Add(gens, Objectify(type, [i, u, j, P]));
od;
od;
od;
fi;
return gens;
end);
# you can't do this operation on arbitrary subsemigroup of Rees matrix
# semigroups since they don't have to be simple and so don't have to have rows,
# columns etc.
# generators for the subsemigroup generated by <IxUxJ>, when <R> is a Rees
# matrix semigroup (and not only a subsemigroup).
InstallMethod(GeneratorsOfReesMatrixSemigroup,
"for a Rees matrix subsemigroup, rows, semigroup, columns",
[IsReesMatrixSubsemigroup, IsList, IsSemigroup, IsList],
function(R, I, U, J)
if not IsReesMatrixSemigroup(R) then
ErrorNoReturn("the first argument must be a Rees matrix semigroup");
elif not IsSubset(Rows(R), I) or IsEmpty(I) then
ErrorNoReturn("the second argument must be a non-empty subset of the ",
"rows of the first argument (a Rees matrix semigroup)");
elif not IsSubsemigroup(UnderlyingSemigroup(R), U) then
ErrorNoReturn("the third argument must be a subsemigroup of the ",
"underlying semigroup of the first argument (a Rees matrix ",
"semigroup)");
elif not IsSubset(Columns(R), J) or IsEmpty(J) then
ErrorNoReturn("the fourth argument must be a non-empty subset of the ",
"columns of the first argument (a Rees matrix semigroup)");
fi;
return GeneratorsOfReesMatrixSemigroupNC(R, I, U, J);
end);
# you can't do this operation on arbitrary subsemigroup of Rees matrix
# semigroups since they don't have to be simple and so don't have to have rows,
# columns etc.
# generators for the subsemigroup generated by <IxUxJ>, when <R> is a Rees
# matrix semigroup (and not only a subsemigroup).
InstallMethod(GeneratorsOfReesZeroMatrixSemigroup,
"for a Rees 0-matrix semigroup, rows, semigroup, columns",
[IsReesZeroMatrixSubsemigroup, IsList, IsSemigroup, IsList],
function(R, I, U, J)
if not IsReesZeroMatrixSemigroup(R) then
ErrorNoReturn("the first argument must be a Rees 0-matrix semigroup");
elif not IsSubset(Rows(R), I) or IsEmpty(I) then
ErrorNoReturn("the second argument must be a non-empty subset of the ",
"rows of the first argument (a Rees 0-matrix semigroup)");
elif not IsSubsemigroup(UnderlyingSemigroup(R), U) then
ErrorNoReturn("the third argument must be a subsemigroup of the ",
"underlying semigroup of the first argument (a Rees ",
"0-matrix semigroup)");
elif not IsSubset(Columns(R), J) or IsEmpty(J) then
ErrorNoReturn("the fourth argument must be a non-empty subset of the ",
"columns of the first argument (a Rees 0-matrix semigroup)");
fi;
return GeneratorsOfReesZeroMatrixSemigroupNC(R, I, U, J);
end);
#
InstallMethod(GeneratorsOfSemigroup, "for a Rees matrix semigroup",
[IsReesMatrixSemigroup],
function(R)
return GeneratorsOfReesMatrixSemigroupNC(R, Rows(R), UnderlyingSemigroup(R),
Columns(R));
end);
#
InstallMethod(GeneratorsOfSemigroup, "for a Rees 0-matrix semigroup",
[IsReesZeroMatrixSemigroup],
function(R)
local gens;
gens:=GeneratorsOfReesZeroMatrixSemigroupNC(R, Rows(R),
UnderlyingSemigroup(R), Columns(R));
if ForAll(Rows(R), i-> ForAll(Columns(R), j-> Matrix(R)[j][i]<>0)) then
Add(gens, MultiplicativeZero(R));
fi;
return gens;
end);
# Note that it is possible that the rows and columns of the matrix only contain
# the zero element, if the resulting semigroup were taken to be in
# IsReesMatrixSemigroup then it would belong to IsReesMatrixSemigroup and
# IsReesZeroMatrixSubsemigroup, so that its elements belong to
# IsReesZeroMatrixSemigroupElement but not to IsReesMatrixSemigroupElement
# (since this makes reference to the whole family used to create the
# semigroups). On the other hand, if we simply exclude the 0, then every method
# for IsReesZeroMatrixSemigroup is messed up because we assume that they always
# contain the 0.
#
# Hence we always include the 0 element, even if all the matrix
# entries corresponding to I and J are non-zero.
InstallMethod(ReesZeroMatrixSubsemigroup,
"for a Rees 0-matrix semigroup, rows, semigroup, columns",
[IsReesZeroMatrixSubsemigroup, IsList, IsSemigroup, IsList],
function(R, I, U, J)
if not IsReesZeroMatrixSemigroup(R) then
ErrorNoReturn("the first argument must be a Rees 0-matrix semigroup");
elif not IsSubset(Rows(R), I) or IsEmpty(I) then
ErrorNoReturn("the second argument must be a non-empty subset of the ",
"rows of the first argument (a Rees 0-matrix semigroup)");
elif not IsSubsemigroup(UnderlyingSemigroup(R), U) then
ErrorNoReturn("the third argument must be a subsemigroup of the ",
"underlying semigroup of the first argument (a Rees ",
"0-matrix semigroup)");
elif not IsSubset(Columns(R), J) or IsEmpty(J) then
ErrorNoReturn("the fourth argument must be a non-empty subset of the ",
"columns of the first argument (a Rees 0-matrix semigroup)");
fi;
return ReesZeroMatrixSubsemigroupNC(R, I, U, J);
end);
InstallGlobalFunction(ReesZeroMatrixSubsemigroupNC,
function(R, I, U, J)
local S;
if U=UnderlyingSemigroup(R) and ForAny(Matrix(R){J}{I}, x-> 0 in x) then
S:=Objectify( NewType( FamilyObj(R),
IsReesZeroMatrixSubsemigroup and IsAttributeStoringRep ), rec() );
SetTypeReesMatrixSemigroupElements(S, TypeReesMatrixSemigroupElements(R));
SetMatrix(S, Matrix(R)); SetUnderlyingSemigroup(S, UnderlyingSemigroup(R));
SetRows(S, I); SetColumns(S, J);
SetParentAttr(S, R);
#it might be that all the matrix entries corresponding to I and J are zero
#and so we can't set IsZeroSimpleSemigroup here.
SetMultiplicativeZero(S, MultiplicativeZero(R));
SetIsSimpleSemigroup(S, false);
SetIsReesZeroMatrixSemigroup(S, true);
return S;
fi;
return Semigroup(GeneratorsOfReesZeroMatrixSemigroupNC(R, I, U, J));
end);
#
InstallMethod(ReesMatrixSubsemigroup,
"for a Rees matrix semigroup, rows, semigroup, columns",
[IsReesMatrixSubsemigroup, IsList, IsSemigroup, IsList],
function(R, I, U, J)
if not IsReesMatrixSemigroup(R) then
ErrorNoReturn("the first argument must be a Rees matrix semigroup");
elif not IsSubset(Rows(R), I) or IsEmpty(I) then
ErrorNoReturn("the second argument must be a non-empty subset of the ",
"rows of the first argument (a Rees matrix semigroup)");
elif not IsSubsemigroup(UnderlyingSemigroup(R), U) then
ErrorNoReturn("the third argument must be a subsemigroup of the ",
"underlying semigroup of the first argument (a Rees ",
"matrix semigroup)");
elif not IsSubset(Columns(R), J) or IsEmpty(J) then
ErrorNoReturn("the fourth argument must be a non-empty subset of the ",
"columns of the first argument (a Rees matrix semigroup)");
fi;
return ReesMatrixSubsemigroupNC(R, I, U, J);
end);
#
InstallGlobalFunction(ReesMatrixSubsemigroupNC,
function(R, I, U, J)
local S;
if U=UnderlyingSemigroup(R) then
S:=Objectify( NewType( FamilyObj(R),
IsReesMatrixSubsemigroup and IsAttributeStoringRep ), rec() );
SetTypeReesMatrixSemigroupElements(S, TypeReesMatrixSemigroupElements(R));
SetMatrix(S, Matrix(R)); SetUnderlyingSemigroup(S, UnderlyingSemigroup(R));
SetRows(S, I); SetColumns(S, J);
SetParentAttr(S, R);
if HasIsSimpleSemigroup(R) and IsSimpleSemigroup(R) then
SetIsSimpleSemigroup(S, true);
fi;
SetIsReesMatrixSemigroup(S, true);
SetIsZeroSimpleSemigroup(S, false);
return S;
fi;
return Semigroup(GeneratorsOfReesMatrixSemigroupNC(R, I, U, J));
end);
#
BindGlobal("_InjectionPrincipalFactor",
function(D, constructor)
local map, inv, G, mat, rep, R, L, x, RR, LL, rms, iso, hom, i, j;
map := IsomorphismPermGroup(GroupHClassOfGreensDClass(D));
inv := InverseGeneralMapping(map);
G := Range(map);
mat := [];
rep := Representative(GroupHClassOfGreensDClass(D));
L := List(GreensHClasses(GreensRClassOfElement(Parent(D), rep)),
Representative);
R := List(GreensHClasses(GreensLClassOfElement(Parent(D), rep)),
Representative);
for i in [1 .. Length(L)] do
mat[i] := [];
for j in [1 .. Length(R)] do
x := L[i] * R[j];
if x in D then
mat[i][j] := x ^ map;
else
mat[i][j] := 0;
fi;
od;
od;
RR := EmptyPlist(Length(R));
LL := EmptyPlist(Length(L));
for j in [1 .. Length(R)] do
for i in [1 .. Length(L)] do
if mat[i][j] <> 0 then
RR[j] := ((mat[i][j] ^ -1) ^ inv) * L[i];
break;
fi;
od;
od;
for i in [1 .. Length(L)] do
for j in [1 .. Length(R)] do
if mat[i][j] <> 0 then
LL[i] := R[j] * (mat[i][j] ^ -1) ^ inv;
break;
fi;
od;
od;
rms := constructor(G, mat);
iso := function(x)
i := PositionProperty(R, y -> y in GreensRClassOfElement(Parent(D), x));
j := PositionProperty(L, y -> y in GreensLClassOfElement(Parent(D), x));
if i = fail or j = fail then
return fail;
fi;
return Objectify(TypeReesMatrixSemigroupElements(rms),
[i, (rep * RR[i] * x * LL[j]) ^ map, j, mat]);
end;
hom := MappingByFunction(D, rms, iso,
function(x)
if x![1] = 0 then
return fail;
fi;
return R[x![1]] * (x![2] ^ inv) * L[x![3]];
end);
SetIsInjective(hom, true);
SetIsTotal(hom, true);
return hom;
end);
InstallMethod(IsomorphismReesMatrixSemigroup, "for a D-class",
[IsGreensDClass],
function(D)
if Number(D, IsIdempotent) <> Length(GreensHClasses(D)) then
ErrorNoReturn("the argument (a Green's D-class) is not a subsemigroup");
fi;
return _InjectionPrincipalFactor(D, ReesMatrixSemigroup);
end);
InstallMethod(IsomorphismReesMatrixSemigroup,
"for a finite simple", [IsSemigroup],
function(S)
local rep, inj;
if not (IsSimpleSemigroup(S) and IsFinite(S)) then
ErrorNoReturn("the argument must be a finite simple semigroup");
fi;
rep := Representative(S);
inj := _InjectionPrincipalFactor(GreensDClassOfElement(S, rep),
ReesMatrixSemigroup);
return MagmaIsomorphismByFunctionsNC(S, Range(inj),
x -> x ^ inj,
x -> x ^ InverseGeneralMapping(inj));
end);
InstallMethod(IsomorphismReesZeroMatrixSemigroup,
"for a finite 0-simple", [IsSemigroup],
function(S)
local D, map, inj, inv;
if not (IsZeroSimpleSemigroup(S) and IsFinite(S)) then
ErrorNoReturn("the argument must be a finite 0-simple semigroup");
fi;
D := First(GreensDClasses(S),
x -> not IsMultiplicativeZero(S, Representative(x)));
map := _InjectionPrincipalFactor(D, ReesZeroMatrixSemigroup);
# the below is necessary since map is not defined on the zero of S
inj := function(x)
if x = MultiplicativeZero(S) then
return MultiplicativeZero(Range(map));
fi;
return x ^ map;
end;
inv := function(x)
if x = MultiplicativeZero(Range(map)) then
return MultiplicativeZero(S);
fi;
return x ^ InverseGeneralMapping(map);
end;
return MagmaIsomorphismByFunctionsNC(S,
Range(map),
inj,
inv);
end);
#
InstallMethod(AssociatedReesMatrixSemigroupOfDClass,
"for a Green's D-class of a semigroup",
[IsGreensDClass],
function(D)
if not IsFinite(Parent(D)) then
TryNextMethod();
fi;
if not IsRegularDClass(D) then
ErrorNoReturn("the argument should be a regular D-class");
fi;
if Length(GreensHClasses(D)) = Number(D, IsIdempotent) then
return Range(IsomorphismReesMatrixSemigroup(D));
fi;
return Range(_InjectionPrincipalFactor(D, ReesZeroMatrixSemigroup));
end);
# so that we can find Green's relations etc
InstallMethod(MonoidByAdjoiningIdentity, [IsReesMatrixSubsemigroup],
function(R)
local M;
M:=Monoid(List(GeneratorsOfSemigroup(R), MonoidByAdjoiningIdentityElt));
SetUnderlyingSemigroupOfMonoidByAdjoiningIdentity(M, R);
return M;
end);
# so that we can find Green's relations etc
InstallMethod(MonoidByAdjoiningIdentity, [IsReesZeroMatrixSubsemigroup],
function(R)
local M;
M:=Monoid(List(GeneratorsOfSemigroup(R), MonoidByAdjoiningIdentityElt));
SetUnderlyingSemigroupOfMonoidByAdjoiningIdentity(M, R);
return M;
end);
# the next two methods by Michael Torpey and Thomas Bourne.
InstallMethod(IsomorphismReesZeroMatrixSemigroup,
"for a Rees 0-matrix subsemigroup", [IsReesZeroMatrixSubsemigroup],
function(U)
local V, iso, inv, hom;
if not IsReesZeroMatrixSemigroup(U) then
TryNextMethod();
elif IsWholeFamily(U) then
return MagmaIsomorphismByFunctionsNC(U, U, IdFunc, IdFunc);
fi;
V:=ReesZeroMatrixSemigroup(UnderlyingSemigroup(U),
Matrix(U){Columns(U)}{Rows(U)});
iso := function(u)
if u = MultiplicativeZero(U) then
return MultiplicativeZero(V);
fi;
return RMSElement(V,
Position(Rows(U),u![1]),
u![2],
Position(Columns(U),u![3]));
end;
inv := function(v)
if v = MultiplicativeZero(V) then
return MultiplicativeZero(U);
fi;
return RMSElement(U, Rows(U)[v![1]], v![2], Columns(U)[v![3]]);
end;
return MagmaIsomorphismByFunctionsNC(U, V, iso, inv);
end);
InstallMethod(IsomorphismReesMatrixSemigroup, "for a Rees matrix subsemigroup",
[IsReesMatrixSubsemigroup],
function(U)
local P, V, iso, inv, hom;
if not IsReesMatrixSemigroup(U) then
TryNextMethod();
elif IsWholeFamily(U) then
return MagmaIsomorphismByFunctionsNC(U, U, IdFunc, IdFunc);
fi;
V:=ReesMatrixSemigroup(UnderlyingSemigroup(U),
List(Matrix(U){Columns(U)}, x-> x{Rows(U)}));
#JDM doing Matrix(U){Columns(U)}{Rows(U)} the resulting object does not know
#IsRectangularTable, and doesn't store this after it is calculated.
iso := function(u)
return RMSElement(V, Position(Rows(U),u![1]), u![2],
Position(Columns(U),u![3]));
end;
inv := function(v)
return RMSElement(U, Rows(U)[v![1]], v![2], Columns(U)[v![3]]);
end;
return MagmaIsomorphismByFunctionsNC(U, V, iso, inv);
end);
#EOF