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|
################################################################################
##
#W libnr.gi library near-rings Christof N"obauer
##
#Y Copyright (C)
##
##
## 13.01.00 N!.multiplication ersetzt, PM
##
##############################################################################
##
InstallValue( NEARRING_PATH_NAME,
DirectoriesPackageLibrary( "sonata", "nr" ) );
InstallValue( NEARRING_WITH_IDENTITY_PATH_NAME,
DirectoriesPackageLibrary( "sonata", "nri" ) );
BindGlobal( "AutoReadNR",
function ( filename )
Read( Filename( NEARRING_PATH_NAME, Concatenation(filename,".nr") ) );
end );
BindGlobal( "AutoReadNRI",
function ( filename )
Read( Filename( NEARRING_WITH_IDENTITY_PATH_NAME, Concatenation(filename,".nr") ) );
end );
AUTO( AutoReadNR, "nr_2-7" , "NR_C2", "NR_C3", "NR_C4", "NR_V4", "NR_C5",
"NR_C6", "NR_S3", "NR_C7" );
AUTO( AutoReadNR, "nr8_1" , "NR_C8" );
AUTO( AutoReadNR, "nr8_2" , "NR_C2xC4" );
AUTO( AutoReadNR, "nr8_3" , "NR_C2xC2xC2" );
AUTO( AutoReadNR, "nr8_4" , "NR_D8" );
AUTO( AutoReadNR, "nr8_5" , "NR_Q8" );
AUTO( AutoReadNR, "nr9_1" , "NR_C9" );
AUTO( AutoReadNR, "nr9_2" , "NR_C3xC3" );
AUTO( AutoReadNR, "nr10_1" , "NR_C10" );
AUTO( AutoReadNR, "nr10_2" , "NR_D10" );
AUTO( AutoReadNR, "nr11_1" , "NR_C11" );
AUTO( AutoReadNR, "nr12_1" , "NR_C12" );
AUTO( AutoReadNR, "nr12_2" , "NR_C2xC6" );
AUTO( AutoReadNR, "nr12_3.1" , "NR_D12_1" );
AUTO( AutoReadNR, "nr12_3.2" , "NR_D12_2" );
AUTO( AutoReadNR, "nr12_3.3" , "NR_D12_3" );
AUTO( AutoReadNR, "nr12_3.4" , "NR_D12_4" );
AUTO( AutoReadNR, "nr12_3.5" , "NR_D12_5" );
AUTO( AutoReadNR, "nr12_3.6" , "NR_D12_6" );
AUTO( AutoReadNR, "nr12_3.7" , "NR_D12_7" );
AUTO( AutoReadNR, "nr12_3.8" , "NR_D12_8" );
AUTO( AutoReadNR, "nr12_3.9" , "NR_D12_9" );
AUTO( AutoReadNR, "nr12_4" , "NR_A4" );
AUTO( AutoReadNR, "nr12_5" , "NR_T" );
AUTO( AutoReadNR, "nr13_1" , "NR_C13" );
AUTO( AutoReadNR, "nr14_1" , "NR_C14" );
AUTO( AutoReadNR, "nr14_2" , "NR_D14" );
AUTO( AutoReadNR, "nr15_1" , "NR_C15" );
########################################################
AUTO( AutoReadNRI, "nri12_3" , "NI12_3" );
AUTO( AutoReadNRI, "nri16_1" , "NI16_1" );
AUTO( AutoReadNRI, "nri16_2" , "NI16_2" );
AUTO( AutoReadNRI, "nri16_3" , "NI16_3" );
AUTO( AutoReadNRI, "nri16_4" , "NI16_4" );
AUTO( AutoReadNRI, "nri16_5" , "NI16_5" );
AUTO( AutoReadNRI, "nri16_6" , "NI16_6" );
AUTO( AutoReadNRI, "nri16_7" , "NI16_7" );
AUTO( AutoReadNRI, "nri16_9" , "NI16_9" );
AUTO( AutoReadNRI, "nri16_10" , "NI16_10" );
AUTO( AutoReadNRI, "nri16_11" , "NI16_11" );
AUTO( AutoReadNRI, "nri17_1" , "NI17_1" );
AUTO( AutoReadNRI, "nri18_1" , "NI18_1" );
AUTO( AutoReadNRI, "nri18_2" , "NI18_2" );
AUTO( AutoReadNRI, "nri18_3" , "NI18_3" );
AUTO( AutoReadNRI, "nri19_1" , "NI19_1" );
AUTO( AutoReadNRI, "nri20_1" , "NI20_1" );
AUTO( AutoReadNRI, "nri20_2" , "NI20_2" );
AUTO( AutoReadNRI, "nri20_3" , "NI20_3" );
AUTO( AutoReadNRI, "nri21_1" , "NI21_1" );
AUTO( AutoReadNRI, "nri22_1" , "NI22_1" );
AUTO( AutoReadNRI, "nri23_1" , "NI23_1" );
AUTO( AutoReadNRI, "nri24_1" , "NI24_1" );
AUTO( AutoReadNRI, "nri24_2" , "NI24_2" );
AUTO( AutoReadNRI, "nri24_3" , "NI24_3" );
AUTO( AutoReadNRI, "nri24_4" , "NI24_4" );
AUTO( AutoReadNRI, "nri24_5" , "NI24_5" );
AUTO( AutoReadNRI, "nri24_7" , "NI24_7" );
AUTO( AutoReadNRI, "nri24_9" , "NI24_9" );
AUTO( AutoReadNRI, "nri25_1" , "NI25_1" );
AUTO( AutoReadNRI, "nri25_2" , "NI25_2" );
AUTO( AutoReadNRI, "nri26_1" , "NI26_1" );
AUTO( AutoReadNRI, "nri27_1" , "NI27_1" );
AUTO( AutoReadNRI, "nri27_2" , "NI27_2" );
AUTO( AutoReadNRI, "nri27_3" , "NI27_3" );
AUTO( AutoReadNRI, "nri27_4" , "NI27_4" );
AUTO( AutoReadNRI, "nri27_5" , "NI27_5" );
AUTO( AutoReadNRI, "nri28_1" , "NI28_1" );
AUTO( AutoReadNRI, "nri28_2" , "NI28_2" );
AUTO( AutoReadNRI, "nri28_3" , "NI28_3" );
AUTO( AutoReadNRI, "nri29_1" , "NI29_1" );
AUTO( AutoReadNRI, "nri30_1" , "NI30_1" );
AUTO( AutoReadNRI, "nri31_1" , "NI31_1" );
############################################################################
##
#F NumberLibraryNearRings
##
NumberLibraryNearRings := function( group )
local grpname_number;
grpname_number := [ [1,1] , 1,
[2,1] , 3,
[3,1] , 5,
[4,1] , 12,
[4,2] , 23,
[5,1] , 10,
[6,1] , 60,
[6,2] , 39,
[7,1] , 24,
[8,1] , 135,
[8,2] , 1159,
[8,3] , 834,
[8,4] , 1447,
[8,5] , 281,
[9,1] , 222,
[9,2] , 264,
[10,1] , 329,
[10,2] , 206,
[11,1] , 139,
[12,1] , 1749,
[12,2] , 3501,
[12,3] , 48137,
[12,4] , 483,
[12,5] , 824,
[13,1] , 454,
[14,1] , 2716,
[14,2] , 1821,
[15,1] , 3817 ];
if Size( group ) < 16 then
return grpname_number[ Position( grpname_number, IdTWGroup( group ) ) + 1 ];
else
Error( "group not recognized" );
fi;
end;
############################################################################
##
#F NumberLibraryNearRingsWithOne
##
NumberLibraryNearRingsWithOne := function( group )
local grpname_number, none;
none := [ [6,2], [8,5], [10,2], [12,4], [12,5], [14,2], [16,8],
[16,12], [16,13], [16,14], [18,4], [18,5], [20,4], [20,5],
[21,2], [22,2], [24,6], [24,8], [24,10], [24,11], [24,12],
[24,13], [24,14], [24,15], [26,2], [28,4], [30,2], [30,3], [30,4] ];
grpname_number := [ [1,1] , 1,
[2,1] , 1,
[3,1] , 1,
[4,1] , 1,
[4,2] , 5,
[5,1] , 1,
[6,1] , 1,
[7,1] , 1,
[8,1] , 1,
[8,2] , 10,
[8,3] , 35,
[8,4] , 7,
[9,1] , 1,
[9,2] , 10,
[10,1] , 1,
[11,1] , 1,
[12,1] , 1,
[12,2] , 9,
[12,3] , 1,
[13,1] , 1,
[14,1] , 1,
[15,1] , 1,
[16,1] , 1,
[16,2] , 37,
[16,3] , 51,
[16,4] , 470,
[16,5] , 2798,
[16,6] , 708,
[16,7] , 4,
[16,9] , 132,
[16,10], 40,
[16,11], 33,
[17,1] , 1,
[18,1] , 1,
[18,2] , 17,
[18,3] , 8,
[19,1] , 1,
[20,1] , 1,
[20,2] , 9,
[20,3] , 1,
[21,1] , 1,
[22,1] , 1,
[23,1] , 1,
[24,1] , 1,
[24,2] , 14,
[24,3] , 136,
[24,4] , 10,
[24,5] , 8,
[24,7] , 7,
[24,9] , 1,
[25,1] , 1,
[25,2] , 16,
[26,1] , 1,
[27,1] , 1,
[27,2] , 20,
[27,3] , 202,
[27,4] , 22,
[27,5] , 4,
[28,1] , 1,
[28,2] , 9,
[28,3] , 1,
[29,1] , 1,
[30,1] , 1,
[31,1] , 1 ];
if Size( group ) < 32 then
if IdTWGroup( group ) in none then
return 0;
else
return grpname_number[ Position( grpname_number, IdTWGroup( group ) ) + 1 ];
fi;
else
Error( "group not recognized" );
fi;
end;
############################################################################
##
#F AllLibraryNearRings
##
AllLibraryNearRings := function( group )
local infolevel, lnrs, tmp;
tmp := LibraryNearRing( group, 1 ); # print a warning?
infolevel := InfoLevel(InfoWarning); # remember InfoLevel
SetInfoLevel( InfoWarning, 0 ); # no warnings
lnrs := List( [1..NumberLibraryNearRings( group )], i ->
LibraryNearRing( group, i ) );
SetInfoLevel( InfoWarning, infolevel ); # reset InfoLevel
return lnrs;
end;
############################################################################
##
#F AllLibraryNearRingsWithOne
##
AllLibraryNearRingsWithOne := function( group )
local position_list, tmp, lnrs, infolevel;
position_list := [
[2,1] , [ 2 ],
[3,1] , [ 3 ],
[4,1] , [ 8 ],
[4,2] , [ 12, 13, 15, 16, 22 ],
[5,1] , [ 7 ],
[6,1] , [ 28 ],
[6,2] , [ ],
[7,1] , [ 19 ],
[8,1] , [ 95 ],
[8,2] , [ 814, 815, 840, 849, 857, 970, 971, 974, 975, 1121 ],
[8,3] , [ 634, 638, 639, 640, 644, 645, 647, 648, 657, 658,
660, 661, 668, 669, 672, 674, 677, 678, 693, 694,
696, 698, 707, 708, 709, 711, 713, 714, 715, 717,
718, 796, 801, 812, 833 ],
[8,4] , [ 842, 844, 848, 849, 1094, 1096, 1097 ],
[8,5] , [ ],
[9,1] , [ 185 ],
[9,2] , [ 208, 209, 211, 215, 216, 219, 222, 225, 226, 255 ],
[10,1] , [ 168 ],
[10,2] , [ ],
[11,1] , [ 130 ],
[12,1] , [ 1117 ],
[12,2] , [ 2096, 2105, 2118, 2120, 2180, 2250, 2255, 2355, 2952 ],
[12,4] , [ ],
[12,5] , [ ],
[13,1] , [ 429 ],
[14,1] , [ 1632 ],
[14,2] , [ ],
[15,1] , [ 2618 ] ];
if Size( group ) < 16 and IdTWGroup( group ) <> [12,3] then
##PM: has to be changed to obtain near-rings in category IsNearRingWithOne
return AllLibraryNearRings( group ){position_list[ Position( position_list, IdTWGroup( group ) ) + 1 ]};
else
if NumberLibraryNearRingsWithOne( group ) = 0 then return []; fi;
tmp := LibraryNearRingWithOne( group, 1 ); # print a warning?
infolevel := InfoLevel(InfoWarning); # remember InfoLevel
SetInfoLevel( InfoWarning, 0 ); # no warnings
lnrs := List( [1..NumberLibraryNearRingsWithOne( group )], i ->
LibraryNearRingWithOne( group, i ) );
SetInfoLevel( InfoWarning, infolevel ); # reset InfoLevel
return lnrs;
fi;
end;
#############################################################################
##
#M LibraryNearRing
## This function 'extracts' a nearring from the nearring library files.
##
InstallMethod(
LibraryNearRing,
"small groups",
true,
[IsGroup, IsInt and IsPosRat],
0,
function( group, num )
local idtw;
# check the arguments
if Size(group) > 15 then
Error( "Usage: LibraryNearRing( <group>, <num> ) where <group> must be ",
"\na group from GroupList of order ",
"less or equal to 15 and \n<num> must be a positive ",
"integer which determines an isomorphism class" );
fi;
Info( InfoWarning, 1, "using isomorphic copy of the group" );
idtw := IdTWGroup( group );
return LibraryNearRing( TWGroup(idtw[1],idtw[2]), num );
end );
InstallMethod(
LibraryNearRing,
"TW groups",
true,
[IsGroup and HasName, IsInt and IsPosRat],
10,
function( group, num )
local n, # help var: a nearring
clmax, # the maximal number of equivalence classes of nearrings
NR, # the nearring to be returned
elms, # help var: the elements of G
i, # help var: a loop variable
tfle, # help var: the record that holds the tfl's of the group endos
f, # help var: a valid function that represents a class of nr's
vf,endos,g,a,a_inv,h,compute_all, ns,
mul, # local function: the multiplication of the nearring
gens, # generators of group
imgs; # images of gens under endomorphisms
# check the arguments
if Size(group) > 15 then
Error( "Usage: LibraryNearRing( <group>, <num> ) where <group> must be ",
"\na group from GroupList of order ",
"less or equal to 15 and \n<num> must be a positive ",
"integer which determines an isomorphism class" );
fi;
n := rec(); n.classes := rec();
if ( Name( group ) = "2/1" ) then
n := NR_C2; ns := "2/1";
elif ( Name( group ) = "3/1" ) then
n := NR_C3; ns := "3/1";
elif ( Name( group ) = "4/1" ) then
n := NR_C4; ns := "4/1";
elif ( Name( group ) = "4/2" ) then
n := NR_V4; ns := "4/2";
elif ( Name( group ) = "5/1" ) then
n := NR_C5; ns := "5/1";
elif ( Name( group ) = "6/1" ) then
n := NR_C6; ns := "6/1";
elif ( Name( group ) = "6/2" ) then
n := NR_S3; ns := "6/2";
elif ( Name( group ) = "7/1" ) then
n := NR_C7; ns := "7/1";
elif ( Name( group ) = "8/1" ) then
n := NR_C8; ns := "8/1";
elif ( Name( group ) = "8/2" ) then
n := NR_C2xC4; ns := "8/2";
elif ( Name( group ) = "8/3" ) then
n := NR_C2xC2xC2; ns := "8/3";
elif ( Name( group ) = "8/4" ) then
n := NR_D8; ns := "8/4";
elif ( Name( group ) = "8/5" ) then
n := NR_Q8; ns := "8/5";
elif ( Name( group ) = "9/1" ) then
n := NR_C9; ns := "9/1";
elif ( Name( group ) = "9/2" ) then
n := NR_C3xC3; ns := "9/2";
elif ( Name( group ) = "10/1" ) then
n := NR_C10; ns := "10/1";
elif ( Name( group ) = "10/2" ) then
n := NR_D10; ns := "10/2";
elif ( Name( group ) = "11/1" ) then
n := NR_C11; ns := "11/1";
elif ( Name( group ) = "12/1" ) then
n := NR_C12; ns := "12/1";
elif ( Name( group ) = "12/2" ) then
n := NR_C2xC6; ns := "12/2";
elif ( Name( group ) = "12/3" ) then
if num in [1..5000] then
n := NR_D12_1;
elif num in [5001..10000] then
n := NR_D12_2;
elif num in [10001..15000] then
n := NR_D12_3;
elif num in [15001..20000] then
n := NR_D12_4;
elif num in [20001..25000] then
n := NR_D12_5;
elif num in [25001..30000] then
n := NR_D12_6;
elif num in [30001..35000] then
n := NR_D12_7;
elif num in [35001..40000] then
n := NR_D12_8;
elif num in [40001..48137] then
n := NR_D12_9;
fi;
ns := "12/3";
elif ( Name( group ) = "12/4" ) then
n := NR_A4; ns := "12/4";
elif ( Name( group ) = "12/5" ) then
n := NR_T; ns := "12/5";
elif ( Name( group ) = "13/1" ) then
n := NR_C13; ns := "13/1";
elif ( Name( group ) = "14/1" ) then
n := NR_C14; ns := "14/1";
elif ( Name( group ) = "14/2" ) then
n := NR_D14; ns := "14/2";
elif ( Name( group ) = "15/1" ) then
n := NR_C15; ns := "15/1";
else
TryNextMethod();
# Print( "There is no group name '", group,
# "' in the nearrings library.\n" );
# return;
fi;
clmax := NumberLibraryNearRings( group );
if num > clmax then
Print( "There are only ", clmax, " isomorphism classes of nearrings ",
"on the group ", group, ".\n" );
return;
fi;
# put the group of the nearring together and define a few help variables
elms := AsSSortedList( group );
tfle := n!.group_endomorphisms;
f := n!.classes!.(num)!.phi;
group!.phi := f;
group!.a_y_i_nrs := n!.classes!.(num)!.autos_yielding_iso_nrs;
# retrieve the group endomorphisms from the Nearrings record
if not HasEndomorphisms( group ) then
# convert the endomorphism record into a list of endomorphisms
i := 1; gens := GeneratorsOfGroup( group ); endos := [];
while IsBound( tfle.(i) ) do
imgs := List( gens, gen -> elms[tfle.(i)[Position(elms,gen)]] );
Add( endos, GroupGeneralMappingByImages( group, group, gens, imgs ) );
i := i + 1;
od;
SetEndomorphisms( group, endos );
fi;
# define a LEFT distributive multiplication
mul := function( y, x )
return elms[ tfle!.(f[ Position( elms, y ) ]) [ Position( elms, x ) ] ];
end;
# put the nearring together
NR := ExplicitMultiplicationNearRingNC( group, mul );
SetLibraryNearRingFlag( NR, true );
SetIdLibraryNearRing( NR, [ group, num ] );
SetName( NR, Concatenation( "LibraryNearRing(", Name( group ), ", ",
String( num ), ")" ) );
return NR;
end );
#############################################################################
##
#M LibraryNearRingWithOne
## This function 'extracts' a nearring from the nearring library files.
##
InstallMethod(
LibraryNearRingWithOne,
"small groups",
true,
[IsGroup, IsInt and IsPosRat],
0,
function( group, num )
local idtw;
# check the arguments
if Size(group) > 32 then
Error( "Usage: LibraryNearRing( <group>, <num> ) where <group> must be ",
"\na group from GroupList of order ",
"less or equal to 15 and \n<num> must be a positive ",
"integer which determines an isomorphism class" );
fi;
Info( InfoWarning, 1, "using isomorphic copy of the group" );
idtw := IdTWGroup( group );
return LibraryNearRingWithOne( TWGroup(idtw[1],idtw[2]), num );
end );
InstallMethod(
LibraryNearRingWithOne,
"TW groups",
true,
[IsGroup and HasName, IsInt and IsPosRat],
10,
function( group, num )
local n, # help var: a nearring
clmax, # the maximal number of equivalence classes of nearrings
NR, # the nearring to be returned
elms, # help var: the elements of G
i, # help var: a loop variable
tfle, # help var: the record that holds the tfl's of the group endos
f, # help var: a valid function that represents a class of nr's
vf,endos,g,a,a_inv,h,compute_all, ns,
mul, # local function: the multiplication of the nearring
gens, # generators of group
imgs, # images of gens under endomorphisms
id; # identity
if Size( group ) < 16 and Name( group ) <> "12/3" then
if num > NumberLibraryNearRingsWithOne( group ) then
Print( "There are only ", NumberLibraryNearRingsWithOne( group ),
" isomorphism classes of nearrings with One ",
"on the group ", group, ".\n" );
return;
else
##PM: has to be changed to obtain near-rings in category IsNearRingWithOne
return AllLibraryNearRingsWithOne( group )[ num ];
fi;
fi;
if ( Name( group ) = "12/3" ) then
n := NI12_3; ns := "12/3";
elif ( Name( group ) = "16/1" ) then
n := NI16_1; ns := "16/1";
elif ( Name( group ) = "16/2" ) then
n := NI16_2; ns := "16/2";
elif ( Name( group ) = "16/3" ) then
n := NI16_3; ns := "16/3";
elif ( Name( group ) = "16/4" ) then
n := NI16_4; ns := "16/4";
elif ( Name( group ) = "16/5" ) then
n := NI16_5; ns := "16/5";
elif ( Name( group ) = "16/6" ) then
n := NI16_6; ns := "16/6";
elif ( Name( group ) = "16/7" ) then
n := NI16_7; ns := "16/7";
elif ( Name( group ) = "16/9" ) then
n := NI16_9; ns := "16/9";
elif ( Name( group ) = "16/10" ) then
n := NI16_10; ns := "16/10";
elif ( Name( group ) = "16/11" ) then
n := NI16_11; ns := "16/11";
elif ( Name( group ) = "17/1" ) then
n := NI17_1; ns := "17/1";
elif ( Name( group ) = "18/1" ) then
n := NI18_1; ns := "18/1";
elif ( Name( group ) = "18/2" ) then
n := NI18_2; ns := "18/2";
elif ( Name( group ) = "18/3" ) then
n := NI18_3; ns := "18/3";
elif ( Name( group ) = "19/1" ) then
n := NI19_1; ns := "19/1";
elif ( Name( group ) = "20/1" ) then
n := NI20_1; ns := "20/1";
elif ( Name( group ) = "20/2" ) then
n := NI20_2; ns := "20/2";
elif ( Name( group ) = "20/3" ) then
n := NI20_3; ns := "20/3";
elif ( Name( group ) = "21/1" ) then
n := NI21_1; ns := "21/1";
elif ( Name( group ) = "22/1" ) then
n := NI22_1; ns := "22/1";
elif ( Name( group ) = "23/1" ) then
n := NI23_1; ns := "23/1";
elif ( Name( group ) = "24/1" ) then
n := NI24_1; ns := "24/1";
elif ( Name( group ) = "24/2" ) then
n := NI24_2; ns := "24/2";
elif ( Name( group ) = "24/3" ) then
n := NI24_3; ns := "24/3";
elif ( Name( group ) = "24/4" ) then
n := NI24_4; ns := "24/4";
elif ( Name( group ) = "24/5" ) then
n := NI24_5; ns := "24/5";
elif ( Name( group ) = "24/7" ) then
n := NI24_7; ns := "24/7";
elif ( Name( group ) = "24/9" ) then
n := NI24_9; ns := "24/9";
elif ( Name( group ) = "25/1" ) then
n := NI25_1; ns := "25/1";
elif ( Name( group ) = "25/2" ) then
n := NI25_2; ns := "25/2";
elif ( Name( group ) = "26/1" ) then
n := NI26_1; ns := "26/1";
elif ( Name( group ) = "27/1" ) then
n := NI27_1; ns := "27/1";
elif ( Name( group ) = "27/2" ) then
n := NI27_2; ns := "27/2";
elif ( Name( group ) = "27/3" ) then
n := NI27_3; ns := "27/3";
elif ( Name( group ) = "27/4" ) then
n := NI27_4; ns := "27/4";
elif ( Name( group ) = "27/5" ) then
n := NI27_5; ns := "27/5";
elif ( Name( group ) = "28/1" ) then
n := NI28_1; ns := "28/1";
elif ( Name( group ) = "28/2" ) then
n := NI28_2; ns := "28/2";
elif ( Name( group ) = "28/3" ) then
n := NI28_3; ns := "28/3";
elif ( Name( group ) = "29/1" ) then
n := NI29_1; ns := "29/1";
elif ( Name( group ) = "30/1" ) then
n := NI30_1; ns := "30/1";
elif ( Name( group ) = "31/1" ) then
n := NI31_1; ns := "31/1";
else
Print( "There is no group name '", group,
"' in the nearrings library.\n" );
return;
fi;
clmax := Length( RecNames( n.classes ) );
if num > clmax then
Print( "There are only ", clmax, " isomorphism classes of nearrings with One",
"on the group ", group, ".\n" );
return;
fi;
# put the group of the nearring together and define a few help variables
elms := AsSSortedList( group );
tfle := n!.group_endomorphisms;
f := n!.classes!.(num)!.phi;
group!.phi := f;
group!.a_y_i_nrs := n!.classes!.(num)!.autos_yielding_iso_nrs;
# retrieve the group endomorphisms from the Nearrings record
if not HasEndomorphisms( group ) then
if not ( Name( group ) in [ "16/5", "27/3" ] ) then
# convert the endomorphism record into a list of endomorphisms
i := 1; gens := GeneratorsOfGroup( group ); endos := [];
while IsBound( tfle.(i) ) do
imgs := List( gens, gen -> elms[tfle.(i)[Position(elms,gen)]] );
Add( endos, GroupGeneralMappingByImages( group, group, gens, imgs ) );
i := i + 1;
od;
SetEndomorphisms( group, endos );
fi;
fi;
# define a LEFT distributive multiplication
mul := function( y, x )
return elms[ tfle!.(f[ Position( elms, y ) ]) [ Position( elms, x ) ] ];
end;
# put the nearring together
##PM: has to be changed to obtain near-rings in category IsNearRingWithOne
NR := ExplicitMultiplicationNearRingNC( group, mul );
SetLibraryNearRingFlag( NR, true );
SetIdLibraryNearRingWithOne( NR, [ group, num ] );
SetName( NR, Concatenation( "LibraryNearRingWithOne(", Name( group ), ", ",
String( num ), ")" ) );
id := One(NR);
return NR;
end );
#############################################################################
##
#F LibraryNearRingInfo( <name>, <list> ). . . . info about library nearrings
##
LibraryNearRingInfo := function( arg )
local N, elms, grpelms, n,
symbols, help, i, k,
name, list, string,
letters, addGroup, PRINTLISTOFSTRINGS;
if not ( Length( arg ) in [ 2, 3 ]
and ( IsString( arg[1] ) or IsGroup( arg[1] ) )
and IsList( arg[2] )
and ForAll( arg[2], l -> IsInt( l ) ) ) then
Error( "Usage: LibraryNearRingInfo( <name>, <list> ) where <name>",
" must be a\ngroup name or a group from _GroupList_ of order le 15 ",
" and <list> must be\na list of numbers",
" of classes" );
fi;
PRINTLISTOFSTRINGS := function ( list )
local i;
Print( "[ ",String(list[1]) );
for i in [2..Length(list)] do
Print( ", ",String(list[i]) );
od;
Print( " ]");
end;
name := arg[1]; list := arg[2];
if IsBound( arg[3] ) then letters := arg[3]; else letters := ""; fi;
if 'C' in letters or 'c' in letters then
Print( "A ... abstract affine\n" );
Print( "B ... boolean\n" );
Print( "C ... commutative\n" );
Print( "D ... distributive\n" );
Print( "F ... nearfield\n" );
Print( "G ... distributively generated\n" );
Print( "I ... integral\n" );
Print( "N ... nilpotent\n" );
Print( "O ... planar\n" );
Print( "P ... prime\n" );
Print( "Q ... quasiregular\n" );
Print( "R ... regular\n" );
Print( "W ... without non-zero nilpotent elements\n" );
Print( "---------------------------------------",
"---------------------------------------\n" );
fi;
n := Size( name );
if n >= 16 then
N := LibraryNearRingWithOne( name, list[ 1 ] );
else
N := LibraryNearRing( name, list[ 1 ] );
fi;
addGroup := GroupReduct(N);
elms := AsList( N );
grpelms := List( elms, GroupElementRepOfNearRingElement );
symbols := Symbols(N);
Print( "---------------------------------------",
"---------------------------------------" );
Print( "\n>>> GROUP: ", Name(addGroup), "\nelements: " );
PRINTLISTOFSTRINGS( symbols ); Print( "\n" );
Print( "\naddition table:\n" );
PrintTable( N, "ea" );
if not ( Name( addGroup ) in [ "16/5", "27/3" ] ) then
Print( "\ngroup endomorphisms:\n" );
for i in [1..Length( Endomorphisms(addGroup) )] do
if i < 10 then
Print( i, ": " );
else
Print( i, ": " );
fi;
PRINTLISTOFSTRINGS( List( grpelms, x -> symbols[ Position( grpelms,
Image( Endomorphisms( addGroup )[i], x ) ) ] ) );
Print( "\n" );
od;
fi;
Print( "\nNEARRINGS:\n" );
Print( "---------------------------------------",
"---------------------------------------" );
for k in list do
if n >= 16 then
N := LibraryNearRingWithOne( name, k );
else
N := LibraryNearRing( name, k );
fi;
elms := AsList( N );
grpelms := List( elms, GroupElementRepOfNearRingElement );
Print( "\n\n", k, ") phi: ", addGroup!.phi, "; " );
for i in addGroup!.a_y_i_nrs do Print( i, ";" ); od;
string := [];
if IsAbstractAffineNearRing( N ) then Add( string, 'A' );
else Add( string, '-' ); fi;
if IsBooleanNearRing( N ) then Add( string, 'B' );
else Add( string, '-' ); fi;
if IsCommutative( N ) then Add( string, 'C' );
else Add( string, '-' ); fi;
if IsDistributiveNearRing( N ) then Add( string, 'D' );
else Add( string, '-' ); fi;
if IsNearField( N ) then Add( string, 'F' );
else Add( string, '-' ); fi;
if IsDgNearRing( N ) then Add( string, 'G' );
else Add( string, '-' ); fi;
if IsIntegralNearRing( N ) then Add( string, 'I' );
else Add( string, '-' ); fi;
if IsNilpotentNearRing( N ) then Add( string, 'N' );
else Add( string, '-' ); fi;
if IsPlanarNearRing( N ) then Add( string, 'O' );
else Add( string, '-' ); fi;
if IsPrimeNearRing( N ) then Add( string, 'P' );
else Add( string, '-' ); fi;
if IsQuasiregularNearRing( N ) then Add( string, 'Q' );
else Add( string, '-' ); fi;
if IsRegularNearRing( N ) then Add( string, 'R' );
else Add( string, '-' ); fi;
if IsNilpotentFreeNearRing( N ) then Add( string, 'W' );
else Add( string, '-' ); fi;
Print( " ", string );
if One( N ) <> fail then Print( "; I = ",
symbols[ Position( elms, One(N) ) ], "\n" );
else
Print( "\n" );
fi;
if 'M' in letters or 'm' in letters then
Print("\nmultiplication table:\n");
PrintTable( N, "m" );
fi;
if 'I' in letters or 'i' in letters then
Print( "\nideals:\n" );
help := AsSSortedList( NearRingIdeals( N ) );
for i in [1..Length(help)] do
Print( i,". " );
PRINTLISTOFSTRINGS(
List( AsSSortedList( help[i] ),
elm -> symbols[Position(elms,elm)] ) );
Print( "\n" );
od;
fi;
if 'L' in letters or 'l' in letters then
Print( "\nleft ideals:\n" );
help := AsSSortedList( NearRingLeftIdeals( N ) );
for i in [1..Length(help)] do
Print( i,". " );
PRINTLISTOFSTRINGS(
List( AsSSortedList( help[i] ),
elm -> symbols[Position(elms,elm)] ) );
Print( "\n" );
od;
fi;
if 'R' in letters or 'r' in letters then
Print( "\nright ideals:\n" );
help := AsSSortedList( NearRingRightIdeals( N ) );
for i in [1..Length(help)] do
Print( i,". " );
PRINTLISTOFSTRINGS(
List( AsSSortedList( help[i] ),
elm -> symbols[Position(elms,elm)] ) );
Print( "\n" );
od;
fi;
if 'V' in letters or 'v' in letters then
Print( "\ninvariant subnearrings:\n" );
help := InvariantSubNearRings( N );
for i in [1..Length(help)] do
Print( i,". " );
PRINTLISTOFSTRINGS(
List( help[i], elm -> symbols[Position(elms,elm)] ) );
Print( "\n" );
od;
fi;
if 'S' in letters or 's' in letters then
Print( "\nsubnearrings:\n" );
help := SubNearRings( N );
for i in [1..Length(help)] do
Print( i,". ");
PRINTLISTOFSTRINGS(
List( help[i], elm -> symbols[Position(elms,elm)] ) );
Print( "\n" );
od;
fi;
if 'E' in letters or 'e' in letters then
Print( "\nnearring endomorphisms: " );
help := Endomorphisms( N );
for i in List( help, e-> Position(Endomorphisms(addGroup), e )) do
Print( i, "; " );
od; Print( "\n" );
fi;
if 'A' in letters or 'a' in letters then
Print( "\nnearring automorphisms: " );
help := Automorphisms( N );
for i in List( help, e -> Position(Endomorphisms(addGroup), e )) do
Print( i, "; " );
od; Print( "\n" );
fi;
Print( "\n---------------------------------------",
"---------------------------------------" );
od;
Print( "\n" );
return;
end;
#####################################################################
##
#F SetSymbolsSupervised( <N>, <L> ) use the symbols in the list <L>
## when printing operation tables
##
InstallGlobalFunction(
SetSymbolsSupervised,
function ( nr, symb )
local symbols;
if not IsHomogeneousList( symb ) then
Error("<symb> must be a list without holes");
elif not ForAll( symb, IsString ) then
Error("<symb> must be a list of strings");
elif Length(Set(symb)) < Length(symb) then
Error("<symb> must be a list without repetitions");
fi;
if Length(symb) < Size(nr) then
Print("Warning: not enough symbols ...",
"extending list automatically\n");
symbols := Symbols(nr);
symbols := Concatenation( symb,
symbols{[Length(symb)+1..Length(symbols)]}
);
else
if Length(symb) > Size(nr) then
Print("Warning: too many symbols ...",
"ignoring the last ",Length(symb)-Size(nr)," symbols\n");
fi;
symbols := symb{[1..Size(nr)]};
fi;
SetSymbols( nr, symbols );
return;
end );
#####################################################################
##
#M Symbols
##
InstallMethod(
Symbols,
"standard",
true,
[IsNearRing],
0,
function ( nr )
return List( [1..Size(nr)], i ->
String( Concatenation( "n", String(i-1) ) ) );
end );
#############################################################################
##
#M MagicNumber
##
BindGlobal( "MAGIC_SUM",
function ( base, list )
local sum, i, limit;
sum := 0; limit := Length(list);
for i in [1..limit] do
sum := sum + base^(limit-i)*list[i];
od;
return sum;
end );
InstallMethod(
MagicNumber,
"default",
true,
[IsNearRing],
0,
function ( nr )
local elms, ml, zeros, mml, e, zero;
elms := AsSSortedList( nr ); zero := elms[1]-elms[1];
ml := []; zeros := [];
for e in elms do
mml := List( elms, f -> f * e);
Add( zeros, Number( mml, x -> x = zero ) );
mml := List( elms, e -> Number( mml, x -> x = e ) );
Sort( mml );
Add( ml, mml );
od;
Sort( zeros );
Sort( ml );
return [zeros, ml];
end );
#############################################################################
##
#M IdLibraryNearRing
##
InstallMethod(
IdLibraryNearRing,
"nearring with One",
true,
[IsNearRing and IsNearRingWithOne],
10,
function ( nr )
local position_list, wrongNumber, groupName, positions;
if Size( nr ) > 15 then
Error( "Only nearrings up to order 15 are in the library" );
fi;
position_list := [
"2/1" , [ 2 ],
"3/1" , [ 3 ],
"4/1" , [ 8 ],
"4/2" , [ 12, 13, 15, 16, 22 ],
"5/1" , [ 7 ],
"6/1" , [ 28 ],
"6/2" , [ ],
"7/1" , [ 19 ],
"8/1" , [ 95 ],
"8/2" , [ 814, 815, 840, 849, 857, 970, 971, 974, 975, 1121 ],
"8/3" , [ 634, 638, 639, 640, 644, 645, 647, 648, 657, 658,
660, 661, 668, 669, 672, 674, 677, 678, 693, 694,
696, 698, 707, 708, 709, 711, 713, 714, 715, 717,
718, 796, 801, 812, 833 ],
"8/4" , [ 842, 844, 848, 849, 1094, 1096, 1097 ],
"8/5" , [ ],
"9/1" , [ 185 ],
"9/2" , [ 208, 209, 211, 215, 216, 219, 222, 225, 226, 255 ],
"10/1" , [ 168 ],
"10/2" , [ ],
"11/1" , [ 130 ],
"12/1" , [ 1117 ],
"12/2" , [ 2096, 2105, 2118, 2120, 2180, 2250, 2255, 2355, 2952 ],
"12/3" , [ 37984 ],
"12/4" , [ ],
"12/5" , [ ],
"13/1" , [ 429 ],
"14/1" , [ 1632 ],
"14/2" , [ ],
"15/1" , [ 2618 ] ];
wrongNumber := IdLibraryNearRingWithOne( nr );
groupName := Name( wrongNumber[1] );
positions := position_list[ Position( position_list, groupName ) + 1 ];
return [ wrongNumber[1], positions[wrongNumber[2]] ];
end );
#############################################################################
##
#M IdLibraryNearRing
##
InstallMethod(
IdLibraryNearRing,
"many classes",
true,
[IsNearRing],
2,
function ( nr )
local addGroup, TWNumber, isoGroup, iso, autos, isos, n, isonr, candidates,
nrCandidates, i, numbers, magicNumber;
if Size( nr ) > 15 then
Error( "Only nearrings up to order 15 are in the library" );
else
# find the isomorphic additive group
addGroup := GroupReduct( nr );
TWNumber := IdTWGroup( addGroup );
isoGroup := TWGroup( TWNumber[1], TWNumber[2] );
if NumberLibraryNearRings( isoGroup ) < 1500 then
TryNextMethod();
fi;
candidates := []; numbers := [];
magicNumber := MagicNumber( nr );
iso := IsomorphismGroups( addGroup, isoGroup );
autos := Automorphisms( isoGroup );
isos := List( autos, aut -> iso * aut );
# filter out possible candidates
# possible candidates have similar multiplication table statistics
for n in [1..NumberLibraryNearRings( isoGroup )] do
isonr := LibraryNearRing( isoGroup, n );
if MagicNumber( isonr ) = magicNumber then
for iso in isos do
if IsMultiplicationRespectingHomomorphism( iso, nr, isonr ) then
return [ isoGroup, n ];
fi;
od;
fi;
od;
fi;
end );
InstallMethod(
IdLibraryNearRing,
"default",
true,
[IsNearRing],
0,
function ( nr )
local addGroup, TWNumber, isoGroup, iso, autos, isos, n, isonr,
nrCandidates, i, numbers, magicNumber;
if Size( nr ) > 15 then
Error( "Only nearrings up to order 15 are in the library" );
elif Size( nr ) = 1 then
return [ TWGroup(1,1), 1 ];
else
# find the isomorphic additive group
addGroup := GroupReduct( nr );
TWNumber := IdTWGroup( addGroup );
isoGroup := TWGroup( TWNumber[1], TWNumber[2] );
numbers := [];
magicNumber := MagicNumber( nr );
# filter out possible candidates
# possible candidates have similar multiplication table statistics
for n in [1..NumberLibraryNearRings( isoGroup )] do
isonr := LibraryNearRing( isoGroup, n );
if MagicNumber( isonr ) = magicNumber then
Add( numbers, n );
fi;
od;
nrCandidates := Length( numbers );
# if there is only 1 candidate, nothing has to be checked
if nrCandidates=1 then
return [isoGroup, numbers[1]];
fi;
# otherwise for all group isomorphisms it has to be checked
# if one of them is a nerring isomorphism
iso := IsomorphismGroups( addGroup, isoGroup );
autos := Automorphisms( isoGroup );
isos := List( autos, aut -> iso * aut );
for i in [1..nrCandidates-1] do
isonr := LibraryNearRing( isoGroup, numbers[i] );
for iso in isos do
if IsMultiplicationRespectingHomomorphism( iso, nr, isonr ) then
return [ isoGroup, numbers[i] ];
fi;
od;
od;
# if the right nearring has not occured yet, it is the last one
return [isoGroup, numbers[nrCandidates]];
fi;
end );
#############################################################################
##
#M IdLibraryNearRingWithOne
##
InstallMethod(
IdLibraryNearRingWithOne,
"default",
true,
[IsNearRing],
0,
function ( nr )
local addGroup, TWNumber, isoGroup, iso, autos, isos, n, isonr, candidates,
nrCandidates, i, numbers, magicNumber;
if Size( nr ) > 31 then
Error( "Only nearrings with One up to order 31 are in the library" );
elif Identity( nr ) = fail then
return fail;
else
# find the isomorphic additive group
addGroup := GroupReduct( nr );
TWNumber := IdTWGroup( addGroup );
isoGroup := TWGroup( TWNumber[1], TWNumber[2] );
if TWNumber in [ [16,5], [27,3] ] then
Print( "Sorry, not all nearrings with this additive group are known\n");
return fail;
fi;
candidates := []; numbers := [];
magicNumber := MagicNumber( nr );
# filter out possible candidates
# possible candidates have similar multiplication table statistics
for n in [1..NumberLibraryNearRingsWithOne( isoGroup )] do
isonr := LibraryNearRingWithOne( isoGroup, n );
if MagicNumber( isonr ) = magicNumber then
Add( candidates, isonr );
Add( numbers, n );
fi;
od;
nrCandidates := Length( candidates );
# if there is only 1 candidate, nothing has to be checked
if nrCandidates=1 then
return [isoGroup, numbers[1]];
fi;
# otherwise for all group isomorphisms it has to be checked
# if one of them is a nerring isomorphism
iso := IsomorphismGroups( addGroup, isoGroup );
autos := Automorphisms( isoGroup );
isos := List( autos, aut -> iso * aut );
for i in [1..nrCandidates-1] do
isonr := candidates[i];
for iso in isos do
if IsMultiplicationRespectingHomomorphism( iso, nr, isonr ) then
return [ isoGroup, numbers[i] ];
fi;
od;
od;
# if the right nearring has not occured yet, it is the last one
return [isoGroup, numbers[nrCandidates]];
fi;
end );
#############################################################################
##
#M IsIsomorphicNearRing
##
InstallMethod(
IsIsomorphicNearRing,
"unequal size",
true,
[IsNearRing, IsNearRing],
100,
function ( nr1, nr2 )
if Size( nr1 ) = Size( nr2 ) then
TryNextMethod();
else
return false;
fi;
end );
InstallMethod(
IsIsomorphicNearRing,
"additive groups not isomorphic",
true,
[IsNearRing, IsNearRing],
80,
function ( nr1, nr2 )
if IsIsomorphicGroup( GroupReduct( nr1 ), GroupReduct( nr2 ) ) then
TryNextMethod();
else
return false;
fi;
end );
InstallMethod(
IsIsomorphicNearRing,
"just one One",
true,
[IsNearRing, IsNearRing],
70,
function ( nr1, nr2 )
if ( Identity( nr1 ) = fail and Identity( nr2 ) = fail ) or
( Identity( nr1 ) <> fail and Identity( nr2 ) <> fail ) then
TryNextMethod();
else
return false;
fi;
end );
InstallMethod(
IsIsomorphicNearRing,
"unequal magic numbers",
true,
[IsNearRing, IsNearRing],
50,
function ( nr1, nr2 )
if ( Size( nr1 ) > 31 ) or
( Size(nr1) > 15 and Identity( nr1 ) = fail ) then
TryNextMethod();
fi;
if MagicNumber( nr1 ) = MagicNumber( nr2 ) then
if Identity( nr1 ) = fail then
return IdLibraryNearRing( nr1 ) = IdLibraryNearRing( nr2 );
else
return IdLibraryNearRingWithOne( nr1 ) = IdLibraryNearRingWithOne( nr2 );
fi;
else
return false;
fi;
end );
InstallMethod(
IsIsomorphicNearRing,
"default",
true,
[IsNearRing, IsNearRing],
0,
function ( nr1, nr2 )
local iso, autgrp;
iso := IsomorphismGroups( GroupReduct(nr1), GroupReduct(nr2) );
autgrp := AutomorphismGroup( GroupReduct(nr2) );
return ForAny( autgrp,
aut -> IsMultiplicationRespectingHomomorphism( iso*aut, nr1, nr2 ) );
end);
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